Measure Data Problems for a Class of Elliptic Equations with Mixed Absorption-Reaction

Marie-Françoise Bidaut-Véron; Marta Garcia-Huidobro; Laurent Véron

doi:10.1515/ans-2021-2124

Open Access Published by De Gruyter March 18, 2021

Measure Data Problems for a Class of Elliptic Equations with Mixed Absorption-Reaction

Marie-Françoise Bidaut-Véron , Marta Garcia-Huidobro and Laurent Véron

From the journal Advanced Nonlinear Studies

https://doi.org/10.1515/ans-2021-2124

Abstract

In the present paper, we study the existence of nonnegative solutions to the Dirichlet problem ℒp,qM⁢u:=-Δ⁢u+up-M⁢|∇⁡u|q=μ in a domain Ω⊂ℝN where μ is a nonnegative Radon measure, when p>1, q>1 and M≥0. We also give conditions under which nonnegative solutions of ℒp,qM⁢u=0 in Ω∖K, where K is a compact subset of Ω, can be extended as a solution of the same equation in Ω.

Keywords: Elliptic Equations; Singularities; Bessel Capacities; Riesz Potential; Maximal Functions

MSC 2010: 35J62; 35J66; 31C15; 28A12

1 Introduction

Let Ω be a bounded domain of ℝN, N≥2, and let ℒp,qM be the operator

(1.1) u ↦ ℒ p , q M ⁢ u := - Δ ⁢ u + | u | p - 1 ⁢ u - M ⁢ | ∇ ⁡ u | q for all ⁢ u ∈ C 2 ⁢ ( Ω ) ,

where M≥0 and p,q>1. We first provide an a priori estimate for a positive solution of (1.1) and its gradient in the range 1<q<p. Then we study under what conditions on the parameters any solution of

ℒ p , q M ⁢ u = 0 in ⁢ Ω ∖ K ,

where K is a compact subset of Ω, can be extended as a solution of the same equation in whole Ω, and if it is the case, whether the solution is bounded or not in Ω. We also consider the Dirichlet problem with measure data

(1.2)

ℒ p , q M ⁢ u = μ in ⁢ Ω ,

u = 0 in ⁢ ∂ ⁡ Ω ,

where μ is a nonnegative bounded Radon measure in Ω and exhibit conditions which guarantee the existence of nonnegative solutions to this problem.

If M=0, then ℒp,qM reduces to the Emden–Fowler operator

u ↦ ℒ p ⁢ u := - Δ ⁢ u + | u | p - 1 ⁢ u .

Singularity problems for solutions of ℒp⁢u=0 have been investigated since forty years, starting with the work of Brezis and Véron [13] who gave conditions for the removability of an isolated singularity. Later on Baras and Pierre [4] extended the result in [13] to more general removable sets, introducing the good framework. They obtained a necessary and sufficient condition expressed in terms of the Bessel capacities cap2,p′ℝN (p′=pp-1) both for the removability of compact subsets of Ω and the solvability of the associated Dirichlet problem with measure data. Another class of operator strongly related to ℒp,qM is the Riccati operator

u ↦ ℛ q M ⁢ u := - Δ ⁢ u - M ⁢ | ∇ ⁡ u | q .

The Dirichlet problem with measure data

- Δ ⁢ u - M ⁢ | ∇ ⁡ u | q = μ in ⁢ Ω ,

u = 0 on ⁢ ∂ ⁡ Ω

has been studied by Maz’ya and Verbitsky [19] and Hansson, Maz’ya and Verbitsky [16] when q>2 (and also in ℝN when q>1) and Phuc [23]. Their results necessitate an extensive use of Riesz potentials.

When M>0, there is a balance between the absorption term |u|p-1⁢u and the source term M⁢|∇⁡u|q, and this interaction is the origin of many unexpected new effects. In the study of singularity problems the effect of the diffusion can be neglectable compared to the two nonlinear terms. The scale of the two opposed reaction terms depends upon the position of q with respect to 2⁢pp+1. This is due to the fact that if q=2⁢pp+1, then (1.1) is equivariant with respect to the scaling transformation Tℓ defined for ℓ>0 by

T ℓ ⁢ [ u ] ⁢ ( x ) = ℓ 2 p - 1 ⁢ u ⁢ ( ℓ ⁢ x ) = ℓ 2 - q q - 1 ⁢ u ⁢ ( ℓ ⁢ x ) .

If q<2⁢pp+1, the absorption term is dominant and the behavior of the singular solutions is modelled by the equation ℒp⁢u=0 studied in [27]. If q>2⁢pp+1, the diffusion is negligible and the behavior of the singular solutions is modelled by positive separable solutions of ℰp,qM⁢u=0, where ℰp,qM is an eikonal-type operator defined by

(1.3) ℰ p , q M ⁢ u = u p - M ⁢ | ∇ ⁡ u | q .

This problem is studied in the forthcoming article [8]. If q=2⁢pp+1, the coefficient M>0 plays a fundamental role in the properties of the set of solutions, in particular for the existence of singular solutions and removable singularities; this is not the case when q≠2⁢pp+1 since by an homothety M can be assumed to be equal to 1.

Brezis and Véron proved in [13] that isolated singularities of solutions of ℒp⁢u=0 are removable when p≥NN-2. The removability property has been extended to more general sets using a capacity framework in [4]. Using a change of variable inspired by [7], where boundary singularities of solutions of (1.3) are considered we prove a series of removability results for solutions of

(1.4) ℒ p , q M ⁢ u = 0 .

Theorem 1.1.

Assume that 0∈Ω⊂RN, N≥3, M>0, p≥NN-2, 1<q≤2⁢pp+1 and (p,q)≠(NN-2,NN-1). Then any nonnegative solution u∈C2⁢(Ω∖{0}) of (1.4) in Ω∖{0} belongs to Wloc1,q⁢(Ω)∩Llocp⁢(Ω), and it can be extended as a weak solution of (1.4) in Ω. Furthermore, if we assume

either p ≥ N N - 2 and 1 < q < 2 ⁢ p p + 1 ,
or p > N N - 2 , q=2⁢pp+1 and

(1.5) M < m * := ( p + 1 ) ⁢ ( ( N - 2 ) ⁢ p - N 2 ⁢ p )

then u∈C2⁢(Ω).

The existence of radial singular solutions when (p,q)=(NN-2,NN-1) and M>0, or when p>NN-2, q=2⁢pp+1 and M≥m* shows the optimality of the statements (see [8]). A series of pointwise a priori estimates concerning u and ∇⁡u are presented in the first section. They are obtained by a combination of Keller–Osserman-type estimates, rescaling techniques and Bernstein method. They play a key role for analyzing the case p=NN-2 in the previous theorem, and will be of fundamental importance in the forthcoming paper [8].

The method introduced in the proof of Theorem 1.1 combined with the result of [4] yields a more general removability result. For such a task we denote by capk,bℝN the Bessel capacity relative to ℝN with order k>0 and power b∈(1,∞). If k∈ℕ* it coincides with the Sobolev capacity associated to the space Wk,b⁢(ℝN) by Calderon’s theorem (see, e.g., [1] for a detailed presentation).

Theorem 1.2.

Let p>NN-2 and NN-2<r<p. Suppose that one of the following conditions is verified:

either q = 2 ⁢ p p + 1 and

0 < M < m * ⁢ ( r ) := ( p + 1 ) ⁢ ( p - r p ⁢ ( r - 1 ) ) p p + 1 ,
or 1 < q < 2 ⁢ p p + 1 and M > 0 .

Then, if K is a compact subset of Ω such that cap2,r′RN⁡(K)=0, any nonnegative solution u∈C2⁢(Ω∖K) of (1.4) in Ω∖K can be extended to Ω as a solution still denoted by u in the sense of distributions in Ω. Furthermore, if r≤2⁢NN-2, then u∈C2⁢(Ω).

Next we obtain sufficient conditions on a positive measure in Ω in order (1.2) be solvable. In the sequel we assume that Ω⊂ℝN, N≥2, is a bounded smooth domain. We denote by 𝔐⁢(Ω) (respectively 𝔐b⁢(Ω)) the set of Radon measures (respectively bounded Radon measures) in Ω and by 𝔐+⁢(Ω) (respectively 𝔐+b⁢(Ω)) its positive cone. The total variation norm of a bounded measure μ is ∥μ∥𝔐.

Since for any μ∈𝔐+b⁢(Ω) the nonnegative solutions of ℒp⁢v=μ and ℛqM⁢w=μ are respectively a subsolution and a supersolution of equation (1.2) and they satisfy 0≤v≤w, the construction of v and w is the key-stone for solving (1.2). It is known that these two problems can be solved when the measure μ satisfies some continuity properties with respect to some specific capacities.

Theorem 1.3.

Assume that p>1, 1<q<2. Let μ∈M+b⁢(Ω). If μ satisfies

(1.6) μ ⁢ ( E ) ≤ C ⁢ min ⁡ { cap 2 , p ′ ℝ N ⁡ ( E ) , cap 1 , q ′ ℝ N ⁡ ( E ) } for all Borel sets ⁢ E ⊂ Ω ,

there is a constant c0>0 such that for any 0≤c≤c0 there exists a function u∈W01,q⁢(Ω)∩Lp⁢(Ω), u≥0, satisfying

(1.7) - ∫ Ω u ⁢ Δ ⁢ ζ ⁢ 𝑑 x + ∫ Ω ( u p - M ⁢ | ∇ ⁡ u | q ) ⁢ ζ ⁢ 𝑑 x = c ⁢ ∫ Ω ζ ⁢ 𝑑 μ for all ⁢ ζ ∈ C c 2 ⁢ ( Ω ¯ ) .

The condition on the measure is satisfied if W-1,q⁢(Ω)↪W-2,p⁢(Ω), and we prove the following:

Corollary 1.4.

Let N⁢pN+p≤q<2 and μ∈M+b⁢(Ω) be such that

(1.8) μ ⁢ ( E ) ≤ C ⁢ cap 1 , q ′ ℝ N ⁡ ( E ) for all Borel sets ⁢ E ⊂ Ω ,

for some C>0. Then there exists a constant c1>0 such that for any 0≤c≤c1 there exists a nonnegative function u∈W01,q⁢(Ω)∩Lp⁢(Ω) satisfying (1.7).

By comparison results between capacities we have another type of result:

Corollary 1.5.

Let NN-1≤q≤2⁢pp+1. If μ∈M+b⁢(Ω) satisfies

(1.9) μ ⁢ ( E ) ≤ C ⁢ cap 2 , p ′ ℝ N ⁡ ( E ) for all Borel sets ⁢ E ⊂ Ω ,

for some C>0, then there exists a constant c2>0 such that for any 0≤c≤c2 there exists a nonnegative function u∈W01,q⁢(Ω)∩Lp⁢(Ω) satisfying (1.7).

As an application of the previous results, we prove the following:

Corollary 1.6.

Let p>1, 1<q<2 and μ∈M+b⁢(Ω). There exists a function u∈W01,q⁢(Ω)∩Lp⁢(Ω) solution of (1.7) if one of the following conditions is satisfied:

When p < N N - 2 and q < N N - 1 , if ∥ μ ∥ 𝔐 ≤ c 3 for some c 3 > 0 .
When p < N N - 2 and N N - 1 ≤ q < 2 , if μ satisfies (1.8) for some C>0. In that case there exists c4>0 such that there must hold 0<c<c4 in problem (1.6).
When p ≥ N N - 2 and q < N N - 1 , if ∥ μ ∥ 𝔐 ≤ c 4 * ⁢ M - 1 q - 1 for some c 4 * = c 4 * ⁢ ( N , q , Ω ) > 0 which can be estimated, and if

μ ⁢ ( E ) = 0 for all Borel sets E ⊂ Ω such that ⁢ cap 2 , p ′ ℝ N ⁡ ( E ) = 0 .

In case (i) we show in a forthcoming article [8] and by a completely different method that there is no restriction on c if μ=c⁢δa for some a∈Ω. In the above mentioned article we construct many types of local or global singular solutions using methods inherited from dynamical systems.

2 Removable Singularities

Throughout this article we denote by c and C generic constants the value of which may vary from one occurrence to another even within a single string of estimates, and by cj (j=1,2,…) some constants which have a more important significance and a more precise dependence with respect to the parameters.

2.1 A Priori Estimates

We give two estimates for positive solutions of (1.1) which differ according to the sign of M. If G is an open subset of ℝN, we set dG⁢(x)=dist⁡(x,∂⁡G)

Proposition 2.1.

Let G⊂RN be an open subset, M>0 and 1<q<p. If u∈C1⁢(G) is a nonnegative solution of (1.1), there holds

(2.1) u ⁢ ( x ) ≤ c 5 ⁢ max ⁡ { M 1 p - q ⁢ ( d G ⁢ ( x ) ) - q p - q , ( d G ⁢ ( x ) ) - 2 p - 1 } for all ⁢ x ∈ G ,

for some c5=c5⁢(N,p,q)>0.

Proof.

Following the method of Keller [17] and Osserman [22], we fix x∈G and 0<a<dG⁢(x), and introduce U⁢(z)=λ⁢(a2-|z-x|2)-b for some b>0. Then putting r=|x-z| and U~⁢(r)=U⁢(z), we have in Ba⁢(x)

L ⁢ U ~ = - U ~ ′′ - N - 1 r ⁢ U ~ ′ - M ⁢ | U ~ ′ | q + U ~ p

= λ ( a 2 - r 2 ) - 2 - b [ λ p - 1 ( a 2 - r 2 ) 2 - b ⁢ ( p - 1 ) + 2 b ( N - 2 ( b + 1 ) ) r 2

- 2 N b a 2 - M 2 q b q λ q - 1 r q ( a 2 - r 2 ) 2 + b - q ⁢ ( b + 1 ) ] .

If M>0, the two necessary conditions on b>0 to be fulfilled is order that U~ is a supersolution in B|a|⁢(a) are

2 - b ⁢ ( p - 1 ) ≤ 0 if and only if b⁢(p-1)≥2,
2 + b - q ⁢ ( b + 1 ) ≥ 2 - b ⁢ ( p - 1 ) if and only if b⁢(p-q)≥q.

The above inequalities are satisfied if

b = max ⁡ { 2 p - 1 , q p - q } .

If q>2⁢pp+1, then b=qp-q and

L ⁢ U ~ ≥ λ ⁢ ( a 2 - r 2 ) - 2 ⁢ p - q p - q ⁢ [ λ q - 1 ⁢ ( λ p - q - M ⁢ 2 q ⁢ b q ⁢ ρ q ) ⁢ ( a 2 - r 2 ) 2 ⁢ p - q ⁢ ( p + 1 ) p - q - ( 3 ⁢ b + 1 ) ⁢ N ⁢ a 2 ] .

There exists c51>0 depending on N, p and q such that if we choose

λ = c 5 1 ⁢ max ⁡ { M 1 p - q ⁢ a q p - q , a 2 ⁢ p ⁢ ( q - 1 ) ( p - 1 ) ⁢ ( p - q ) } ,

there holds

(2.2) L ⁢ U ~ ≥ 0 in ⁢ B a ⁢ ( x ) .

Since U~⁢(z)→∞ when r→a, we derive by the maximum principle that u≤U~ in Ba⁢(x). In particular,

u ⁢ ( x ) ≤ U ~ ⁢ ( x ) = λ ⁢ a - 2 ⁢ q p - q = c 5 1 ⁢ max ⁡ { M 1 p - q ⁢ a - q p - q , a - 2 p - 1 } .

If q≤2⁢pp+1, then b=2p-1 and

L U ~ ≥ λ ( | a | 2 - ρ 2 ) - 2 ⁢ p p - 1 [ λ p - 1 + 2 p - 1 ( N - 2 ⁢ ( p + 1 ) p - 1 ) ρ 2 - 2 ⁢ N p - 1 | a | 2

- M 2 q ( 2 p - 1 ) q λ q - 1 ρ q ( | a | 2 - ρ 2 ) 2 ⁢ p - q ⁢ ( p + 1 ) p - 1 ]

≥ λ ⁢ ( | a | 2 - ρ 2 ) - 2 ⁢ p p - 1 ⁢ [ λ p - 1 - c 2 ⁢ | a | 2 - c 3 ⁢ λ q - 1 ⁢ M ⁢ | a | 4 ⁢ p - q ⁢ ( p + 3 ) p - 1 ] .

Hence, if q=2⁢pp+1, then (2.2) holds if for some c52>0 depending on N,p,q,

λ = c 5 2 ⁢ max ⁡ { M p + 1 p ⁢ ( p - 1 ) , 1 } ⁢ | a | 2 p - 1 ,

which yields

u ⁢ ( x ) ≤ U ~ ⁢ ( x ) = λ ⁢ a - 4 p - 1 = c 5 2 ⁢ max ⁡ { M p + 1 p ⁢ ( p - 1 ) , 1 } ⁢ a - 2 p - 1 .

While if q<2⁢pp+1, we choose

λ = c 5 3 ⁢ max ⁡ { M 1 p - q ⁢ a 4 ⁢ p - q ⁢ ( p + 3 ) ( p - 1 ) ⁢ ( p - q ) , a 2 p - 1 } ,

where c53>0=c53⁢(N,p,q), which implies

u ⁢ ( x ) ≤ U ~ ⁢ ( x ) = λ ⁢ a - 4 p - 1 = c 5 3 ⁢ max ⁡ { M 1 p - q ⁢ a - q p - q , a - 2 p - 1 } .

By letting a↑dG⁢(x), we derive (2.1) with a constant c5=c53, depending on N,p,q. ∎

Corollary 2.2.

Under the assumptions of Proposition 2.1 with G=B2⁢R∖{0}, there holds for x∈BR∖{0},

u ⁢ ( x ) ≤ c 5 ⁢ max ⁡ { M 1 p - q ⁢ | x | - q p - q , | x | - 2 p - 1 } .

We infer from Proposition 2.1 an estimate of the gradient of a positive solution when M>0. If we set σ=2⁢p-q⁢(p+1), then σ>0 (respectively σ<0) according to 2⁢p>q⁢(p+1) (respectively 2⁢p<q⁢(p+1)).

Proposition 2.3.

Let p>q>1. For any M0>0 and R>0 there exists a constant c8=c8⁢(N,p,q,M0⁢Rσp-1) such that for 0<M≤M0 there holds:

If q ≤ 2 ⁢ p p + 1 (then σ ≥ 0 ), any positive solution u of ( 1.1 ) in B 2 ⁢ R ∖ { 0 } satisfies

(2.3) | ∇ ⁡ u ⁢ ( x ) | ≤ c 8 ⁢ max ⁡ { M 1 p - q ⁢ | x | - p p - q , | x | - p + 1 p - 1 }

for all x ∈ B R ∖ { 0 } .
If 2 ⁢ p p + 1 ≤ q ≤ 2 (then σ ≤ 0 ), any positive solution u of ( 1.1 ) in ℝ N ∖ B ¯ R 2 satisfies ( 2.1 ) for all x ∈ ℝ N ∖ B R .

Proof.

(i) For 0<r<2⁢R we set

u ⁢ ( x ) = r - 2 p - 1 ⁢ u r ⁢ ( x r ⁢ b ⁢ i ⁢ g ⁢ g ⁢ r ) = r - 2 p - 1 ⁢ u r ⁢ ( y ) with ⁢ y = x r .

If r2<|x|<2⁢r, then 12<|y|<2 and ur>0 satisfies

(2.4) - Δ ⁢ u r + u r p - M ⁢ r 2 ⁢ p - q ⁢ ( p + 1 ) p - 1 ⁢ | ∇ ⁡ u r | q = 0 in ⁢ B 2 ∖ B 1 2 .

Since 0<M⁢rσp-1≤M⁢(2⁢R)σp-1≤M0⁢(2⁢R)σp-1 as σ≥0, it follows that

(2.5) max ⁡ { | ∇ ⁡ u r ⁢ ( z ) | : 2 3 < | z | < 3 2 } ≤ c ⁢ max ⁡ { | u r ⁢ ( z ) | : 1 2 < | z | < 2 } ,

where c depends on N,p,q and Rσp-1⁢M0 (see, e.g., [15, Chapter 13]). From Proposition 2.1 there holds

max ⁡ { | u r ⁢ ( z ) | : 1 2 < | z | ≤ 2 } ≤ 2 2 p - 1 ⁢ c 5 ⁢ max ⁡ { M 1 p - q ⁢ r 2 ⁢ p - q ⁢ ( p + 1 ) ( p - 1 ) ⁢ ( p - q ) , 1 }

by (2.1). Therefore

max ⁡ { | ∇ ⁡ u ⁢ ( y ) | : r 2 < | z | < 2 ⁢ r } ≤ 2 2 p - 1 ⁢ c ⁢ c 5 ⁢ r - p + 1 p - 1 ⁢ max ⁡ { M 1 p - q ⁢ r 2 ⁢ p - q ⁢ ( p + 1 ) ( p - 1 ) ⁢ ( p - q ) , 1 } ≤ c 8 ⁢ max ⁡ { M 1 p - q ⁢ | x | - p p - q , | x | - p + 1 p - 1 } ,

which implies (2.3).

(ii) For r>R we define ur as in (i). It satisfies (2.4) and since σ≤0, we have again

0 < M ⁢ r σ p - 1 ≤ M ⁢ R σ p - 1 ≤ M 0 ⁢ R σ p - 1

if r≥R. Since 1<q<2, it follows that (2.5) holds and we derive (2.3).∎

Remark.

If q=2⁢pp+1, the constant c8 depends only on N and p.

The previous estimate necessitates 1<q≤2. This limitation can be bypassed in some cases using the Bernstein approach.

Lemma 2.4.

Assume that p,q>1 and M>0. If u∈C2⁢(B¯2⁢R) is a nonnegative solution of (1.1) in B2⁢R, there holds

(2.6) | ∇ ⁡ u ⁢ ( x ) | ≤ c 9 ⁢ ( | x | - 1 q - 1 + max | z - x ] ≤ | x | 2 ⁡ u p q ⁢ ( z ) ) for all ⁢ x ∈ B R 2 ,

where c9>0 depends on N, p, q and M.

Proof.

Set z=|∇⁡u|2. Then by a classical computation and the use of Schwarz inequality,

- Δ ⁢ | ∇ ⁡ u | 2 + 1 N ⁢ ( Δ ⁢ u ) 2 + 〈 ∇ ⁡ Δ ⁢ u , ∇ ⁡ u 〉 ≤ 0 .

Replacing Δ⁢u by its expression from ℒp,qM⁢u=0, we obtain

- Δ ⁢ z + 2 N ⁢ ( u 2 ⁢ p + M 2 ⁢ z q - 2 ⁢ M ⁢ u p ⁢ z q 2 ) + 2 ⁢ p ⁢ u p - 1 ⁢ z ≤ q ⁢ M ⁢ z q 2 - 1 ⁢ 〈 ∇ ⁡ z , ∇ ⁡ u 〉 .

We notice that

q ⁢ M ⁢ z q 2 - 1 ⁢ 〈 ∇ ⁡ z , ∇ ⁡ u 〉 ≤ q ⁢ M ⁢ z q 2 - 1 ⁢ | ∇ ⁡ z | ⁢ z = q ⁢ M ⁢ z q 2 ⁢ | ∇ ⁡ z | z ≤ M 2 ⁢ z q 2 ⁢ N + 2 ⁢ N ⁢ q 2 M 2 ⁢ | ∇ ⁡ z | 2 z

and

4 ⁢ M N ⁢ u p ⁢ z q 2 ≤ M 2 ⁢ z q 2 ⁢ N + 8 ⁢ u 2 ⁢ p N ⁢ M 2 ,

thus

- Δ ⁢ z + M 2 ⁢ z q N ≤ 2 ⁢ N ⁢ q 2 M 2 ⁢ | ∇ ⁡ z | 2 z + 2 N ⁢ ( 4 M 2 - 1 ) ⁢ u 2 ⁢ p .

For simplicity we set

A = M 2 N ,

B = 2 ⁢ N ⁢ q 2 M 2 ,

C = 2 N ⁢ ( 4 M 2 - 1 ) + ⁢ max | z - x | ≤ | x | 2 ⁡ u 2 ⁢ p ⁢ ( z ) .

Then z satisfies

- Δ ⁢ z + A ⁢ z q ≤ B ⁢ | ∇ ⁡ z | 2 z + C in ⁢ B R 2 ⁢ ( x )

and we obtain by [5, Lemma 3.1] (see also a simpler approach in [6, Lemma 2.2])

z ⁢ ( x ) ≤ c 10 ⁢ ( | x | - 2 q - 1 + C 1 q ) ,

where c10>0 depends on N, p, q and M. This yields (2.6). ∎

Remark.

The constants c9 and c10 can be expressed in terms of M, but their stability when M→0 is not clear since in the limit case of the equation ℒp⁢u=0 the estimate of the gradient obtained by a very different and much simpler method combining the Keller–Osserman estimate and scaling methods.

Using Corollary 2.2, we obtain the new estimate:

Corollary 2.5.

Assume that 1<q<p and M>0. Then any nonnegative solution u∈C2⁢(B2⁢R) of (1.1) satisfies

(2.7) | ∇ ⁡ u ⁢ ( x ) | ≤ c 11 ⁢ ( | x | - 1 q - 1 + max ⁡ { M p q ( p - q ⁢ | x | - p p - q , | x | - 2 ⁢ p q ⁢ ( p - 1 ) } ) for all ⁢ x ∈ B R 2 ,

where c11>0 depends on N, p, q and M.

Then we can combine this estimate with Proposition 2.1 to complete the cases not treated in Proposition 2.3 and obtain the following:.

Proposition 2.6.

Let 1<q<p. For any M>0 there exists a constant c12=c12⁢(N,p,q,M)>0 such that:

If 2 ⁢ p p + 1 ≤ q < p , any positive solution u of ( 1.1 ) in B 2 ⁢ R ∖ { 0 } with 0 < R ≤ 1 satisfies

(2.8) | ∇ ⁡ u ⁢ ( x ) | ≤ c 12 ⁢ max ⁡ { M p q ⁢ ( p - q ) ⁢ | x | - p p - q , | x | - 2 ⁢ p q ⁢ ( p - 1 ) } for all ⁢ x ∈ B R ∖ { 0 } .
If 1 < q ≤ 2 ⁢ p p + 1 , any positive solution u of ( 1.1 ) in ℝ N ∖ B ¯ R 2 with R ≥ 1 satisfies

(2.9) | ∇ ⁡ u ⁢ ( x ) | ≤ c 12 ⁢ max ⁡ { M p q ⁢ ( p - q ) ⁢ | x | - p p - q , | x | - 1 q - 1 } for all ⁢ x ∈ ℝ N ∖ B R .

Proof.

We can compare the different exponents of |x| which appear in the expressions (2.3) and (2.7)

(2.10)

p p - q < p + 1 p - 1 < 2 ⁢ p q ⁢ ( p - 1 ) < 1 q - 1 if ⁢ 1 < q < 2 ⁢ p p + 1 ,

1 q - 1 < 2 ⁢ p q ⁢ ( p - 1 ) < p + 1 p - 1 < p p - q if ⁢ 2 ⁢ p p + 1 < q < p ,

with equality if q=2⁢pp+1. If 1<q≤2⁢pp+1 (respectively 2⁢pp+1≤2), estimate (2.3) is better than (2.7) in BR∖{0} (respectively ℝN∖B2⁢R). Then (2.8) and (2.9) follow from (2.7) and (2.10). ∎

In the case M<0 an upper estimate on a solution is obtained by combining a result of Lions and the method of Keller and Osserman.

Proposition 2.7.

Let G⊂RN be an open subset, M≤0 and p,q>1. If u∈C1⁢(G) is a nonnegative solution of Lp,qM⁢u=0, there exists c6=c6⁢(N,p)>0, c7=c7⁢(N,q)>0 and δ=δ⁢(G)>0 such that there holds for all x∈G and 0<δ≤δ⁢(G),

(2.11) u ⁢ ( x ) ≤ min ⁡ { c 6 ⁢ ( d G ⁢ ( x ) ) - 2 p - 1 , c 7 ⁢ | M | - 1 q - 1 ⁢ ( d G ⁢ ( x ) ) - 2 - q q - 1 + max d G ⁢ ( z ) = δ ⁡ u ⁢ ( z ) } .

Proof.

This estimate follows from the fact that the solutions of ℒp,qM⁢u=0 are subsolutions of ℒp⁢u=0 and ℛqM⁢u=0. The estimate

u ⁢ ( x ) ≤ c 6 ⁢ ( d G ⁢ ( x ) ) - 2 p - 1

corresponds to the Keller–Osserman estimate for solutions of ℒp⁢u=0. The second estimate corresponds to the fact that if u is a positive solution of ℛqM⁢u=0 in G, there holds (see [18, Theorem IV-1])

| ∇ ⁡ u ⁢ ( x ) | ≤ c 7 ′ ⁢ | M | - 1 q - 1 ⁢ ( d G ⁢ ( x ) ) - 1 q - 1 .

Integrating this inequality yields the second part of the inequality. ∎

Remark.

This estimate can be transformed into the universal estimate

u ⁢ ( x ) ≤ min ⁡ { c 6 ⁢ ( d G ⁢ ( x ) ) - 2 p - 1 , c 7 ⁢ | M | - 1 q - 1 ⁢ ( d G ⁢ ( x ) ) - 2 - q q - 1 + c 6 ⁢ δ - 2 p - 1 } ,

since maxdG⁢(z)=δ⁡u⁢(z)≤c6⁢δ-2p-1 by (2.11).

The gradient estimates are due to Nguyen [20, Proposition 1.1]. Below we recall his result proved by the Bernstein method in a more general framework but which can also be obtained by scaling techniques in the present case.

Proposition 2.8.

Let p>1 and 1<q<2. For any M<0 and R>0 there exists a constant c12′>0, depending on N, p, q, M, such that if u is a positive solution of (1.1) in B2⁢R∖{0}, there holds

u ⁢ ( x ) + | x | ⁢ | ∇ ⁡ u ⁢ ( x ) | ≤ c 12 ′ ⁢ max ⁡ { | x | - 2 p - 1 , | x | - 2 - q q - 1 } for all ⁢ x ∈ B R ∖ { 0 } .

Remark.

There are many estimates of positive solutions of (1.1) (or even with up replaced by f⁢(u)) in a domain which tends to infinity on the boundary (large solutions) or of solutions in ℝN (ground states). Many of these estimates are obtained by comparison with one-dimensional problems and they can be found in [2, 3, 14].

2.2 Proof of Theorem 1.1

Without loss of generality we can assume that u∈C2(Ω¯∖{0} and B¯2⁢R0⊂Ω with 2⁢R0≤1. If M≤0, then u is a nonnegative subsolution of -Δ⁢u+up=0, hence it is bounded in Ω¯ by [13].

Step 1.

We assume M>0 and we prove first that under condition (i) or (ii), |∇⁡u|q∈L1⁢(Ω), u∈Lp⁢(Ω), and then

∫ Ω ( - u ⁢ Δ ⁢ ζ + u p ⁢ ζ - M ⁢ | ∇ ⁡ u | q ⁢ ζ ) ⁢ 𝑑 x = 0 for all ⁢ ζ ∈ W 2 , ∞ ⁢ ( Ω ) ∩ C c 1 ⁢ ( Ω ¯ ) .

By Proposition 2.3

| ∇ ⁡ u | q ≤ c ⁢ | x | - ( p + 1 ) ⁢ q p - 1 in ⁢ B R 0 ,

since q≤2⁢pp+1, and where c depends also on M. By (i) or (ii), (p+1)⁢qp-1<N. Hence ∇⁡u∈Llocq⁢(Ω). For any ϵ>0 small enough denote by ρϵ a nonnegative C∞-function such that supp⁡(ρϵ)⊂B¯ϵ, 0≤ρϵ≤1, |∇⁡ρϵ|≤2⁢ϵ-1⁢χB¯ϵ and we set ηϵ=1-ρϵ. Then

- ∫ B 2 ⁢ R 0 〈 ∇ ⁡ u , ∇ ⁡ ρ ϵ 〉 ⁢ 𝑑 x + ∫ B 2 ⁢ R 0 u p ⁢ η ϵ ⁢ 𝑑 x + ∫ ∂ ⁡ B 2 ⁢ R 0 ∂ ⁡ u ∂ ⁡ 𝐧 ⁢ 𝑑 S = M ⁢ ∫ B 2 ⁢ R 0 | ∇ ⁡ u | q ⁢ η ϵ ⁢ 𝑑 x .

(2.12) | ∫ B 2 ⁢ R 0 〈 ∇ ⁡ u , ∇ ⁡ ρ ϵ 〉 ⁢ 𝑑 x | ≤ 2 ⁢ c N ⁢ ϵ N q ′ - 1 ⁢ ( ∫ B ϵ | ∇ ⁡ u | q ⁢ 𝑑 x ) 1 q → 0 as ⁢ ϵ → 0 ,

since 1<q≤NN-1. Since |∇⁡u|q∈L1⁢(B2⁢R0), we deduce by monotone convergence that up∈L1⁢(B2⁢R0). Finally, if ζ∈C0∞⁢(Ω) and ζϵ=ζ⁢ηϵ, we have

∫ Ω 〈 ∇ ⁡ u , ∇ ⁡ ζ ϵ 〉 ⁢ 𝑑 x + ∫ Ω u p ⁢ ζ ϵ ⁢ 𝑑 x - M ⁢ ∫ Ω | ∇ ⁡ u | q ⁢ ζ ϵ ⁢ 𝑑 x = 0 .

Letting ϵ→0 and using (2.12), we infer that u satisfies

∫ Ω 〈 ∇ ⁡ u , ∇ ⁡ ζ 〉 ⁢ 𝑑 x + ∫ Ω u p ⁢ ζ ⁢ 𝑑 x - M ⁢ ∫ Ω | ∇ ⁡ u | q ⁢ ζ ⁢ 𝑑 x = 0 .

Hence it is a weak solution of (1.4) in Ω.

Step 2.

Let us assume that p>NN-2. If u is nonnegative and not identically zero, it is positive in Ω∖{0} by the maximum principle. We set u=vb with 0<b≤1. Then

- Δ ⁢ v - ( b - 1 ) ⁢ | ∇ ⁡ v | 2 v + 1 b ⁢ v 1 + ( p - 1 ) ⁢ b - M ⁢ b q - 1 ⁢ v ( b - 1 ) ⁢ ( q - 1 ) ⁢ | ∇ ⁡ v | q = 0 .

For ϵ>0,

v ( b - 1 ) ⁢ ( q - 1 ) ⁢ | ∇ ⁡ v | q ≤ q ⁢ ϵ 2 q 2 ⁢ | ∇ ⁡ v | 2 v + 2 - q 2 ⁢ ϵ 2 2 - q ⁢ v 1 + 2 ⁢ b ⁢ ( q - 1 ) 2 - q .

Therefore

(2.13) - Δ ⁢ v + ( 1 - b - M ⁢ q ⁢ b q - 1 ⁢ ϵ 2 q 2 ) ⁢ | ∇ ⁡ v | 2 v + 1 b ⁢ v 1 + b ⁢ ( p - 1 ) - M ⁢ b q - 1 ⁢ 2 - q 2 ⁢ ϵ 2 2 - q ⁢ v 1 + 2 ⁢ b ⁢ ( q - 1 ) 2 - q = 0 .

We notice that 1+2⁢b⁢(q-1)2-q=1+b⁢(p-1)-a with a=b⁢2⁢p-(p+1)⁢q2-q≥0. We fix b as follows:

(2.14) ( p - 1 ) ⁢ b + 1 = N N - 2 ⇔ b = 2 ( N - 2 ) ⁢ ( p - 1 ) ,

hence p>NN-2 if and only if 0<b<1. Next we impose

1 - b - M ⁢ q ⁢ b q - 1 ⁢ ϵ 2 q 2 = 0 ⇔ ϵ = ( 2 ⁢ ( 1 - b ) M ⁢ q ⁢ b q - 1 ) q 2 = ( 2 ⁢ ( ( N - 2 ) ⁢ p - N ) M ⁢ q ⁢ b q - 1 ⁢ ( N - 1 ) ⁢ ( p - 1 ) ) q 2 .

This transforms (2.13) into

(2.15) - Δ ⁢ v + ( N - 2 ) ⁢ ( p - 1 ) 2 ⁢ v N N - 2 - ( 2 - q ) ⁢ b q - 1 2 ⁢ ( q 2 ⁢ ( 1 - b ) ) q 2 - q ⁢ M 2 2 - q ⁢ v N N - 2 - a ≤ 0 .

We first assume that 0<q<2⁢pp+1. Then a>0, hence there exists A>0, depending on M, such that

- Δ ⁢ v + ( N - 2 ) ⁢ ( p - 1 ) 4 ⁢ v N N - 2 ≤ A .

Set v~=(v-c⁢AN-2N)+NN-2 with c=(4(N-2)⁢(p-1))NN-2 satisfies

- Δ ⁢ v ~ + ( N - 2 ) ⁢ ( p - 1 ) 4 ⁢ v ~ N N - 2 ≤ 0 .

By [13], v~≤max∂⁡Ω⁡v~ which implies v≤c⁢AN-2N+max∂⁡Ω⁡v and therefore u⁢(x)≤B for some B≥0 in Ω. Furthermore, |∇⁡u|q-1∈Lqq-1⁢(Ω) since ∇⁡u∈Lq⁢(Ω), and qq-1>N as we assume q<NN-1. Writing (1.4) under the form

- Δ ⁢ u + u p - M ⁢ C ⁢ ( x ) ⁢ | ∇ ⁡ u | = 0

with C⁢(x)=|∇⁡u⁢(x)|q-1, it follows from Serrin’s theorem [25, Theorem 10] that the singularity at 0 is removable and u can be extended as a regular solution of (1.4) in Ω. Hence u∈C2⁢(Ω). Then we assume that q=2⁢pp+1. By the choice of b in (2.14), inequality (2.13) becomes

(2.16) - Δ ⁢ v + ( 1 - b - M ⁢ p ⁢ b p - 1 p + 1 ⁢ ϵ p + 1 p p + 1 ) ⁢ | ∇ ⁡ v | 2 v + ( 1 b - M ⁢ b p - 1 p + 1 ( p + 1 ) ⁢ ϵ p + 1 ) ⁢ v N N - 2 ≤ 0 .

Notice that

(2.17) 1 b - M ⁢ b p - 1 p + 1 ( p + 1 ) ⁢ ϵ p + 1 = 0 ⇔ ϵ = ( M p + 1 ) 1 p + 1 ⁢ b 2 ⁢ p ( p + 1 ) 2 ,

and therefore

(2.18) 1 - b - M ⁢ p ⁢ b p - 1 p + 1 ⁢ ϵ p + 1 p p + 1 = 1 - b - p ⁢ b ⁢ ( M p + 1 ) p + 1 p .

This coefficient vanishes if

p ⁢ ( M p + 1 ) p + 1 p = p ⁢ ( N - 1 ) - ( N + 1 ) 2 .

Therefore, if M satisfies

(2.19) p ⁢ ( M p + 1 ) p + 1 p = p ⁢ ( N - 2 ) - N 2 ,

we can choose ϵ>0 so that the coefficient of v(p-1)⁢b+1 in (2.19) is equal to some τ>0. Therefore v satisfies

- Δ ⁢ v + τ ⁢ v N N - 2 ≤ 0 in ⁢ Ω ∖ { 0 } .

It follows from [13] that v≤max∂⁡Ω⁡v and the same type of uniform estimate holds for u. This ends the case p>NN-2.

Step 3.

Finally, we assume p=NN-2 and 1<q<2⁢pp+1=NN-1. From (2.6),

M ⁢ | ∇ ⁡ u ⁢ ( x ) | q ≤ c 9 ⁢ | x | - q ⁢ p + 1 p - 1 = c 9 ⁢ | x | - q ⁢ ( N - 1 ) := Q ⁢ ( x )

and Q∈L1⁢(B2⁢R0). Let {σn}⊂C0∞⁢(ℝN) such that 0≤σn≤1

σ n ⁢ ( x ) = { 1 if ⁢ 2 n ≤ | x | ≤ R 0 , 0 if ⁢ | x | ∈ [ 0 , 1 n ] ∪ [ 2 ⁢ R 0 , ∞ ) ,

and

| Δ ⁢ σ n | ≤ 2 ⁢ N ⁢ n 2 ⁢ χ B 2 n ∖ B 1 n + ϕ ,

where ϕ is a smooth nonnegative function with support in B2⁢R0∖BR0. Then

(2.20) - ∫ { 1 n ≤ | x | ≤ 2 n } u ⁢ Δ ⁢ σ n ⁢ 𝑑 x - ∫ { R 0 ≤ | x | ≤ 2 R 0 } u ⁢ Δ ⁢ σ n ⁢ 𝑑 x + ∫ 1 n ≤ | x | u p ⁢ σ n ⁢ 𝑑 x = M ⁢ ∫ 1 n ≤ | x | | ∇ ⁡ u | q ⁢ σ n ⁢ 𝑑 x .

The right-hand side of (2.20) is bounded since |∇⁡u|∈Lq⁢(B2⁢R0), the second term on the left is also uniformly bounded. Using the fact that |x|N-2⁢u⁢(x) is bounded by (2.1), we get

| ∫ { 1 n ≤ | x | ≤ 2 n } u ⁢ Δ ⁢ σ n ⁢ 𝑑 x | ≤ C ,

for some C>0 independent of n. Letting n→∞, we infer that u∈Llocp⁢(Ω). By the maximum principle

u ⁢ ( x ) ≤ u 1 ⁢ ( x ) = C ⁢ 𝐆 B 2 ⁢ R 0 ⁢ [ Q ] ⁢ ( x ) + max | z | = 2 ⁢ R 0 ⁡ u ⁢ ( z ) ,

where 𝐆B2⁢R0 denotes the Green kernel in B2⁢R0. Since Q⁢(x)=C⁢|x|-q⁢(N-1), a direct computation shows that u1⁢(x)≤cN⁢C⁢|x|2-q⁢(N-1)=cN⁢C⁢|x|2-N+ϵ for some ϵ>0. We can write (1.1) under the form

- Δ ⁢ u + c ⁢ ( x ) ⁢ u + d ⁢ ( x ) ⁢ | ∇ ⁡ u | = 0 in ⁢ Ω ∖ { 0 } ,

with c⁢(x)=u2N-2 and d⁢(x)=|∇⁡u|q-1. Then we have c∈LN2+ϵ1⁢(B2⁢R0) and d∈LN+ϵ2⁢(B2⁢R0); with ϵ1,ϵ2>0. It follows from [25, Theorem 10] that 0 is a removable singularity for u in the sense that it can be extended as a C2-solution in Ω. Theorem 1.1 is proved. ∎

When the conditions of the theorem are not fulfilled, there exist singular solutions. However, these singular solutions may exhibit different types of behavior according 1<q<2⁢pp+1, 2⁢pp+1<q<2 and q=2⁢pp+1. In this case there may exist radial separable solutions of (1.4) under the form uX⁢(r)=X⁢r-2p-1. By setting α=2p-1, it follows that X satisfies

(2.21) Φ p ⁢ ( X ) := X p - 1 - M ⁢ α 2 ⁢ p p + 1 ⁢ X p - 1 p + 1 + α ⁢ ( N - 2 - α ) = 0

The following result is easy to prove by a standard analysis of the function Φp.

Proposition 2.9.

Let p>1 and M∈R.

If M is arbitrary and 1 < p < N N - 2 , or M > 0 and p = N N - 2 , there exists one and only one positive solution X 1 to ( 2.21 ).
If p > N N - 2 and M > m * , there exist two positive solutions X 1 < X 2 to ( 2.21 ).
If p > N N - 2 and M = m * there exists one positive solution X 1 to ( 2.21 ).
If p > N N - 2 and 0 < M < m * , or M ≤ 0 and p ≥ N N - 2 , there exists no positive solution to ( 2.21 ).

When q≠2⁢pp+1, then the existence of singular solutions is much involved. It is developed in the subsequent paper [8].

Remark.

It is noticeable that in the case q=2⁢pp+1, p>NN-2 and M≥m*, the equation exhibits a phenomenon which is characteristic of Lane–Emden-type equations

- Δ ⁢ u = u p in ⁢ B 1 ∖ { 0 } .

If u is nonnegative, then there exists α≥0 such that

- Δ ⁢ u = u p + α ⁢ δ 0 in ⁢ 𝒟 ′ ⁢ ( B 1 ) .

If 1<p<NN-2, then α can be positive, but if p≥NN-2, then α=0. This means that the singularity cannot be seen in the sense of distributions, however there truly exist singular solutions, e.g., if p>NN-2,

u s ⁢ ( x ) = c N , p ⁢ | x | - 2 p - 1 .

Here also for q=2⁢pp+1, p>NN-2, the isolated singularities are not seen in the sense of distributions.

2.3 Proof of Theorem 1.2

As in the proof of Theorem 1.1, we distinguish according to whether 1<q<2⁢pp+1 or q=2⁢pp+1 holds. Without loss of generality we can suppose that u>0. We perform the same change of unknown as in the previous theorem by putting u=vb, but now we choose b as follows:

( p - 1 ) ⁢ b + 1 = r ⇔ b = r - 1 p - 1 ,

and we first assume that

(2.22) 1 - b - M ⁢ q ⁢ b q - 1 ⁢ ϵ 2 q 2 = 0 ⇔ ϵ = ( 2 ⁢ ( 1 - b ) M ⁢ q ⁢ b q - 1 ) q 2 = ( 2 ⁢ ( p - r ) M ⁢ q ⁢ ( p - 1 ) ⁢ b q - 1 ) q 2 .

Hence (2.15) becomes

(2.23) - Δ ⁢ v + p - 1 r - 1 ⁢ v r - ( 2 - q ) ⁢ b q - 1 2 ⁢ ( q 2 ⁢ ( 1 - b ) ) q 2 - q ⁢ M 2 2 - q ⁢ v ( 2 ⁢ r - p - 1 ) ⁢ q + 2 ⁢ ( p - r ) ( p - 1 ) ⁢ ( 2 - q ) ≤ 0 .

Then

r ≥ ( 2 ⁢ r - p - 1 ) ⁢ q + 2 ⁢ ( p - r ) ( p - 1 ) ⁢ ( 2 - q ) ⇔ 2 ⁢ p - q ⁢ ( p + 1 ) ≤ r ⁢ ( 2 ⁢ p - q ⁢ ( p + 1 ) ) ,

since 1<r<p.

Assuming first that q<2⁢pp+1, we obtain from (2.23)

- Δ ⁢ v + p - 1 2 ⁢ ( r - 1 ) ⁢ v r ≤ A

for some constant A≥0. Since cap2,r′ℝN⁡(K)=0, the function v is bounded from [4] and v≤c⁢A1r+max∂⁡Ω⁡v for some c>0, hence u is also uniformly upper bounded in Ω by some constant a.

Next we have to show that ∇⁡u∈Lq⁢(Ω). Let {ρn} be a sequence of nonnegative C0∞⁢(Ω)-functions such that 0≤ρn≤1, ρn=1 in a small enough neighborhood of K and ∥ρn∥W2,r′→0 when n→∞, and set ηn=1-ρn. Since

∫ Ω u ⁢ Δ ⁢ ρ n ⁢ 𝑑 x - ∫ ∂ ⁡ Ω ∂ ⁡ u ∂ ⁡ 𝐧 ⁢ 𝑑 S + ∫ Ω u p ⁢ η n ⁢ 𝑑 x = M ⁢ ∫ Ω | ∇ ⁡ u | q ⁢ η n ⁢ 𝑑 x

and

| ∫ Ω u ⁢ Δ ⁢ ρ n ⁢ 𝑑 x | ≤ c ⁢ ∥ u ∥ L ∞ ⁢ ∥ ρ n ∥ W 2 , r ′ → 0 as ⁢ n → ∞ ,

we get

∫ Ω u p ⁢ 𝑑 x - ∫ ∂ ⁡ Ω ∂ ⁡ u ∂ ⁡ 𝐧 ⁢ 𝑑 S = M ⁢ ∫ Ω | ∇ ⁡ u | q ⁢ 𝑑 x .

Hence ∇⁡u∈Lq⁢(Ω). If ζ∈C0∞⁢(Ω) and ζn=ζ⁢ηn, there holds

- ∫ Ω η n ⁢ u ⁢ Δ ⁢ ζ ⁢ 𝑑 x + ∫ Ω ζ ⁢ u ⁢ Δ ⁢ ρ n ⁢ 𝑑 x + ∫ Ω u p ⁢ ζ n ⁢ 𝑑 x = M ⁢ ∫ Ω | ∇ ⁡ u | q ⁢ ζ n ⁢ 𝑑 x .

Since the second term on the left-hand side tends to 0 and ζn→ζ when n→∞, we obtain that

- ∫ Ω u ⁢ Δ ⁢ ζ ⁢ 𝑑 x + ∫ Ω u p ⁢ ζ ⁢ 𝑑 x = M ⁢ ∫ Ω | ∇ ⁡ u | q ⁢ ζ ⁢ 𝑑 x .

Hence u is a solution in the sense of distribution in Ω.

Next we show that ∇⁡u∈L2⁢(Ω). Multiplying (1.4) by u⁢ηn and integrating, we obtain

∫ Ω | ∇ ⁡ u | 2 ⁢ η n ⁢ 𝑑 x - ∫ Ω u ⁢ 〈 ∇ ⁡ u , ∇ ⁡ ρ n 〉 ⁢ 𝑑 x - ∫ ∂ ⁡ Ω u ⁢ ∂ ⁡ u ∂ ⁡ 𝐧 ⁢ 𝑑 S + ∫ Ω u p + 1 ⁢ η n ⁢ 𝑑 x = M ⁢ ∫ Ω u ⁢ | ∇ ⁡ u | q ⁢ η n ⁢ 𝑑 x .

∫ Ω u ⁢ 〈 ∇ ⁡ u , ∇ ⁡ ρ n 〉 ⁢ 𝑑 x = 1 2 ⁢ ∫ Ω 〈 ∇ ⁡ u 2 , ∇ ⁡ ρ n 〉 ⁢ 𝑑 x = - 1 2 ⁢ ∫ Ω u 2 ⁢ Δ ⁢ ρ n ⁢ 𝑑 x

and

| ∫ Ω u 2 ⁢ Δ ⁢ ρ n ⁢ 𝑑 x | ≤ c ⁢ ∥ u ∥ L ∞ 2 ⁢ ∥ ρ n ∥ W 2 , r ′ = o ⁢ ( 1 ) as ⁢ n → ∞ ,

we infer that

∫ Ω | ∇ ⁡ u | 2 ⁢ 𝑑 x - ∫ ∂ ⁡ Ω u ⁢ ∂ ⁡ u ∂ ⁡ 𝐧 ⁢ 𝑑 S + ∫ Ω u p + 1 ⁢ 𝑑 x = M ⁢ ∫ Ω u ⁢ | ∇ ⁡ u | q ⁢ 𝑑 x .

Finally, if ζ∈C0∞⁢(Ω) and ζn=ζ⁢ηn, then

∫ Ω η n ⁢ 〈 ∇ ⁡ u , ∇ ⁡ ζ 〉 ⁢ 𝑑 x - ∫ Ω ζ ⁢ 〈 ∇ ⁡ u , ∇ ⁡ ρ n 〉 ⁢ 𝑑 x + ∫ Ω u p ⁢ ζ n ⁢ 𝑑 x = M ⁢ ∫ Ω | ∇ ⁡ u | q ⁢ ζ n ⁢ 𝑑 x .

Since r≤2⁢NN-2, there holds

∥ ρ n ∥ W 1 , 2 ≤ c ⁢ ∥ ρ n ∥ W 2 , r ′ ⟹ ∥ ρ n ∥ W 1 , 2 → 0 as ⁢ n → ∞ .

Using the fact that ∇⁡u∈L2⁢(Ω) and Hölder’s inequality, we derive

∫ Ω ζ ⁢ 〈 ∇ ⁡ u , ∇ ⁡ ρ n 〉 ⁢ 𝑑 x → 0 as ⁢ n → ∞ .

Hence

∫ Ω 〈 ∇ ⁡ u , ∇ ⁡ ζ 〉 ⁢ 𝑑 x + ∫ Ω u p ⁢ ζ ⁢ 𝑑 x = M ⁢ ∫ Ω | ∇ ⁡ u | q ⁢ ζ ⁢ 𝑑 x .

This implies that u is a weak solution of (1.4) and it is therefore C2 in Ω.

Next we assume q=2⁢pp+1. We choose b=r-1p-1 and (2.16) becomes

- Δ ⁢ v + ( 1 - b - M ⁢ p ⁢ b p - 1 p + 1 ⁢ ϵ p + 1 p p + 1 ) ⁢ | ∇ ⁡ v | 2 v + ( 1 b - M ⁢ b p - 1 p + 1 ( p + 1 ) ⁢ ϵ p + 1 ) ⁢ v r ≤ 0 .

If (2.17) holds with this choice of b, (2.18) becomes

1 - b - M ⁢ p ⁢ b p - 1 p + 1 ⁢ ϵ p + 1 p p + 1 = 1 - b - p ⁢ b ⁢ ( M p + 1 ) p + 1 p

= 1 p - 1 ⁢ ( p - r - p ⁢ ( r - 1 ) ⁢ ( M p + 1 ) p + 1 p ) .

If M<mr* defined by (1.5), we can choose ϵ such that

1 - b - M ⁢ p ⁢ b p - 1 p + 1 ⁢ ϵ p + 1 p p + 1 = 0

and

1 b - M ⁢ b p - 1 p + 1 ( p + 1 ) ⁢ ϵ p + 1 = τ := τ ⁢ ( ϵ ) > 0 .

Then v satisfies

- Δ ⁢ v + τ ⁢ v r ≤ 0 in ⁢ Ω ∖ K .

Since cap2,r′ℝN⁡(K)=0, it follows from [4] that v≤maxx∈∂⁡Ω⁡v⁢(x). Hence u is bounded. The different steps of the proof in the first case applies without any modification: first ∇⁡u∈Lq⁢(Ω) and the equation holds in the sense of distributions in Ω, then ∇⁡u∈L2⁢(Ω) and since r≤2⁢NN-2 we infer that u is a weak solution and thus a strong one. ∎

3 Measure Data

Let Ω⊂ℝN be a bounded smooth domain with diameter smaller than 2⁢R. Also any Radon measure in Ω is extended by 0 in Ωc with the same notation.

3.1 Proof of Theorem 1.3: The Case 1<q<NN-1

If 1<q<NN-1, then assumption (1.6) with μ≥0 reduces to

μ ⁢ ( K ) ≤ C ⁢ cap 2 , p ′ ℝ N ⁡ ( K ) for all compact set ⁢ K ⊂ Ω .

The construction of solutions is based upon the following result due to Boccardo, Murat and Puel [10]. It is concerned with a general quasilinear equation in a domain G⊂ℝN:

𝒬 ⁢ ( u ) := - Δ ⁢ u + B ⁢ ( ⋅ , u , ∇ ⁡ u ) = 0 in ⁢ 𝒟 ′ ⁢ ( G ) ,

where B∈C⁢(G×ℝ×ℝN) satisfies

| B ⁢ ( x , r , ξ ) | ≤ Γ ⁢ ( | r | ) ⁢ ( 1 + | ξ | 2 ) for all ⁢ ( x , r , ξ ) ∈ G × ℝ × ℝ N ,

for some continuous increasing function Γ from ℝ+ to ℝ+.

Theorem 3.1.

Let G be a bounded domain in RN. If there exists a supersolution ϕ and a subsolution ψ of the equation Q⁢v=0 belonging to W1,∞⁢(G) and such that ψ≤ϕ, then for any χ∈W1,∞⁢(G) satisfying ψ≤χ≤ϕ there exists a function u∈W1,2⁢(G) solution of Q⁢u=0 such that ψ≤u≤ϕ and u-χ∈W01,2⁢(G).

The sub- and super-solutions are linked to the two problems in which p and q are bigger than 1, and μ and ω are Radon measures

(3.1)

- Δ ⁢ v + | v | p - 1 ⁢ v = μ in ⁢ Ω ,

v = 0 in ⁢ ∂ ⁡ Ω

and

(3.2)

- Δ ⁢ w - M ⁢ | ∇ ⁡ w | q = ω in ⁢ Ω ,

w = 0 in ⁢ ∂ ⁡ Ω .

It is proved in [4, Theorem 4.1] that problem (3.1) admits a solution, v∈Lp⁢(Ω), necessarily unique, if and only if μ is absolutely continuous with respect to the Bessel capacity cap2,p′ℝN, that is:

For any compact set E ⊂ Ω , cap 2 , p ′ ℝ N ⁡ ( E ) = 0 ⟹ | μ | ⁢ ( E ) = 0 .

Concerning (3.2), from [21, Theorem 1.9] a sufficient condition for solvability is the following estimate:

(3.3) For any compact set E ⊂ Ω , | ω | ⁢ ( E ) ≤ C ⁢ cap 1 , q ′ ℝ N ⁡ ( E ) , for some C > 0 .

When μ is nonnegative and has compact support in Ω, this condition turns out to be necessary. If (3.3) is satisfied, there exists ϵ0>0 such that (3.2) admits a solution with ω replaced by ϵ⁢ω with 0<ϵ≤ϵ0. Furthermore, ∇⁡w∈Lq⁢(Ω) and the following estimates hold [9, Theorem 1.2]:

(3.4) | ∇ ⁡ w ⁢ ( x ) | ≤ c 13 ⁢ ϵ ⁢ 𝐈 1 2 ⁢ R ⁢ [ | ω | ] ⁢ ( x ) ,

at least if ω has compact support or is a smooth function, and, with no such conditions on μ,

(3.5) | w ⁢ ( x ) | ≤ c 14 ⁢ ϵ ⁢ 𝐆 Ω ⁢ [ | ω | ] ⁢ ( x ) ,

with c13,c14 depending on N and q, where 𝐈12⁢R is the truncated Riesz potential in ℝN defined for any measure μ by

𝐈 1 2 ⁢ R ⁢ [ μ ] ⁢ ( x ) = ∫ 0 2 ⁢ R μ ⁢ ( B ρ ⁢ ( x ) ) ρ N - 1 ⁢ d ⁢ ρ ρ for all ⁢ x ∈ ℝ N ,

and 𝐆Ω the Green potential in Ω. If R=∞, we denote by 𝐈1:=𝐈1∞ the classical Riesz potential; if Ω=ℝN, the role of 𝐆Ω is played by the Newtonian potential 𝐈2. We start with the following easy result.

Lemma 3.2.

Let r>1, k∈N* and μ∈M+⁢(Ω). If μ∈W-k,r⁢(Ω) is nonnegative, then there exists C>0 such that

μ ⁢ ( E ) ≤ C ⁢ ( cap k , r ′ Ω ⁡ ( E ) ) 1 r ′ for any compact set ⁢ E ⊂ Ω .

where r′=rr-1. Conversely, when k=1,2 and μ satisfies

(3.6) μ ⁢ ( E ) ≤ C ⁢ cap k , r ′ Ω ⁡ ( E ) for all compact set ⁢ E ⊂ Ω ,

for some C>0, then:

if k = 2 , then μ ∈ W - 2 , r ⁢ ( Ω ) ,
if k = 1 , then μ ∈ W - 1 , r ⁢ ( Ω ) .

Proof.

Assume first that μ∈𝔐+⁢(Ω)∩W-k,r⁢(Ω). Let ζ∈C0∞⁢(Ω) such that 0≤ζ≤1 and ζ≥χK. Then

μ ⁢ ( E ) ≤ ∫ Ω ζ ⁢ 𝑑 μ = 〈 μ , ζ 〉 ≤ ∥ ζ ∥ W 0 k , r ′ ⁢ ∥ μ ∥ W - k , r .

By the definition of capacity,

μ ⁢ ( K ) ≤ ∥ μ ∥ W - k , r ⁢ ( cap k , r ⁢ p ′ Ω ⁡ ( K ) ) 1 r ′ .

Conversely, if (3.6) holds with k=2, there exists ϵ0>0 such that for every ϵ∈(0,ϵ0] there exists z∈Lr⁢(Ω) satisfying

- Δ ⁢ z = z r + ϵ ⁢ μ in ⁢ Ω ,

z = 0 on ⁢ ∂ ⁡ Ω

(see [24, Theorem 2.10, Remark 2.11]). Since z≥𝐆Ω⁢[ϵ⁢μ], it follows that 𝐆Ω⁢[μ]∈Lr⁢(Ω) and therefore we have μ∈W-2,r⁢(Ω). Since 𝐆Ω is an isomorphism from Lr′⁢(Ω) into W2,r′⁢(Ω)∩W01,r′⁢(Ω), we infer by duality that 𝐆Ω is an isomorphism from W-2,r⁢(Ω) into Lr⁢(Ω). Hence μ∈W-2,r⁢(Ω).

Finally, if (3.6) holds with k=1, then there exists ϵ0>0 such that for every ϵ∈(0,ϵ0] there exists z∈W1,r⁢(Ω) satisfying

- Δ ⁢ z = | ∇ ⁡ z | r + ϵ ⁢ μ in ⁢ Ω ,

z = 0 on ⁢ ∂ ⁡ Ω .

Then z satisfies z≥𝐆Ω⁢[ϵ⁢μ]. Since z∈Lr*⁢(Ω) by the Sobolev imbedding theorem, we have 𝐆Ω⁢[μ]∈Lr*⁢(Ω), which implies the claim. ∎

Proof of Theorem 3.1.

We put μn=μ∗ηn, where {ηn}⊂C0∞⁢(ℝN) is a sequence of mollifiers with the property that supp⁡(ηn)⊂B1n, and we denote by vn the solution of

- Δ ⁢ v + v p = ϵ ⁢ μ n ⁢ χ Ω in ⁢ Ω ,

v = 0 on ⁢ ∂ ⁡ Ω .

Since μ satisfies (3.6), so does μn with the same constant C. Hence μn∈W-2,p⁢(Ω) and μn→μ in W-2,p⁢(Ω) as n→∞. We also denote by zn a nonnegative solution of

- Δ ⁢ z = z p + ϵ ⁢ μ n ⁢ χ Ω in ⁢ Ω ,

z = 0 on ⁢ ∂ ⁡ Ω

and by wn a nonnegative solution of

- Δ ⁢ w = M ⁢ | ∇ ⁡ w | q + ϵ ⁢ μ n ⁢ χ Ω in ⁢ Ω ,

w = 0 on ⁢ ∂ ⁡ Ω .

Since wn is C2, it is unique by the strong maximum principle. Then there holds by (3.4) and (3.5),

v n ≤ ϵ 𝐆 Ω [ μ n ] ≤ w n ≤ c 14 ϵ 𝐆 Ω [ μ n ] ≤ c 14 ϵ 𝐈 2 [ μ n ] ⌊ Ω ,

| ∇ ⁡ w n | ≤ c 13 ⁢ ϵ ⁢ 𝐈 1 2 ⁢ R ⁢ [ μ n ] .

Since vn and wn are respectively a subsolution and a supersolution of

(3.7)

- Δ ⁢ u + u p = M ⁢ | ∇ ⁡ u | q + ϵ ⁢ μ n in ⁢ Ω ,

u = 0 on ⁢ ∂ ⁡ Ω ,

it follows by Theorem 3.1 that there exists u=un∈W01,∞⁢(Ω) satisfying (3.7) in the sense that for any ζ∈Cc2⁢(Ω¯) there holds

(3.8) - ∫ Ω u n ⁢ Δ ⁢ ζ ⁢ 𝑑 x + ∫ Ω ( u n p - M ⁢ | ∇ ⁡ u n | q ) ⁢ ζ ⁢ 𝑑 x = ϵ ⁢ ∫ Ω ζ ⁢ 𝑑 μ n .

It is unique by the strong maximum principle and it satisfies

(3.9) v n ≤ u n ≤ w n ≤ c 14 ϵ 𝐈 2 [ μ n ] ⌊ Ω .

Since 𝐈2[μn]⌊Ω is uniformly bounded in Lp⁢(Ω), the sequence of functions {un} shares this property. If η=𝐆Ω⁢[1], there holds

∫ Ω ( u n + η ⁢ u n p ) ⁢ 𝑑 x = M ⁢ ∫ Ω | ∇ ⁡ u n | q ⁢ η ⁢ 𝑑 x + ϵ ⁢ ∫ Ω η ⁢ 𝑑 μ n .

Hence |∇⁡un| is uniformly bounded in LdΩq⁢(Ω), where dΩ⁢(x)=dist⁡(x,∂⁡Ω). By (3.4),

(3.10) | ∇ u n | ≤ c 13 ϵ 𝐈 1 2 ⁢ R [ | μ n - u n p | ] ≤ c 13 ( ϵ 𝐈 1 2 ⁢ R [ μ n ] + 𝐈 1 2 ⁢ R [ u n p ] ) ≤ c 13 ( ϵ 𝐈 1 2 ⁢ R [ μ n ] + c 14 p ϵ p 𝐈 1 2 ⁢ R [ ( 𝐈 2 [ μ n ⌊ Ω ] ) p ] ) .

Using [16, Lemma 4.2], we have equivalence between

(3.11) 𝐈 1 [ ( 𝐈 2 [ μ n ⌊ Ω ] ) p ] ≤ c 15 𝐈 1 [ μ n ⌊ Ω ] ,

and

(3.12) 𝐈 2 [ ( 𝐈 2 [ μ n ⌊ Ω ] ) p ] ≤ c 17 𝐈 2 [ μ n ⌊ Ω ] ,

and c17≤c15≤C⁢(N,p)⁢c17. Moreover, since diam⁡(Ω)<2⁢R, it follows that

(3.13) ( 𝐈 1 2 ⁢ R ⁢ [ ( 𝐈 2 ⁢ [ μ ] ) p ] ) q ≤ c 21 q ⁢ ( 𝐈 1 ⁢ [ μ ] ) q ≤ c q ⁢ c 21 q ⁢ ( 𝐈 1 2 ⁢ R ⁢ [ μ ] ) q ,

for some c=c⁢(N,R)>0. Next, it is quoted in [16, Theorem 1.1] that inequality (3.12) is equivalent to the main assumption of Theorem 1.3,

(3.14) μ n ⌊ Ω ( E ) ≤ C cap 2 , p ′ ℝ N ( E ) for all compact set E ⊂ Ω ,

for some C>0. Actually this equivalence is proved in [19]. By [1, Theorem 3.14 (a)] there exists A=A⁢(N)>0 such that

| { x ∈ ℝ N : | 𝐈 1 [ μ n ⌊ Ω ] ( x ) | > λ } | ≤ A λ - N N - 1 ∥ μ n ⌊ Ω ∥ L 1 N N - 1 .

Clearly, the above inequality holds if 𝐈1 is replaced by 𝐈12⁢R and ℝN by Ω. This is an estimate of 𝐈12⁢R[μn⌊Ω] in the Lorentz space LNN-1,∞⁢(Ω) (or Marcinkiewicz space). Clearly,

∥ μ n ⌊ Ω ∥ L 1 ≤ ∥ μ ∥ 𝔐 .

Therefore (3.10) implies that |∇⁡un| is bounded in LNN-1,∞⁢(Ω), hence equi-integrable in Lq⁢(Ω) since q<NN-1. By Lemma 3.2 (ii) and classical harmonic analysis results, 𝐈2[μn⌊Ω→𝐈2[μ⌊Ω in Lp⁢(Ω) (see, e.g., [26]). It follows from (3.9) that un is equi-integrable in Lp⁢(Ω). By standard results on elliptic equations and measure theory [11, Corollary IV], the sequences {un} and {|∇⁡un|} are relatively compact in L1⁢(Ω). Hence there exist a subsequence {nj}, converging to ∞ and a function u∈W01,q⁢(Ω)∩Lp⁢(Ω) such that unj→u in W1,1⁢(Ω) and a.e. in Ω. Since {un} and {|∇⁡un|} are also equi-integrable in Lp⁢(Ω) and Lq⁢(Ω), respectively, we infer that unj→u in W01,q⁢(Ω)∩Lp⁢(Ω). It follows from Vitali’s convergence theorem that unj→u in Lp⁢(Ω) and ∇⁡unj→∇⁡u in Lq⁢(Ω). Letting nj→∞ in (3.8), we conclude that the identity

- ∫ Ω u ⁢ Δ ⁢ ζ ⁢ 𝑑 x + ∫ Ω ( u p - M ⁢ | ∇ ⁡ u | q ) ⁢ ζ ⁢ 𝑑 x = ϵ ⁢ ∫ Ω ζ ⁢ 𝑑 μ n

holds for any ζ∈Cc2⁢(Ω¯). Clearly,

v ≤ u ≤ w ≤ C ϵ 𝐈 2 [ μ ] ⌊ Ω ,

where v and w are respectively the solution of (3.1) and the minimal solution of (3.2) with μ replaced by ϵ⁢μ. ∎

3.2 Proof of Theorem 1.3: The General Case

The approach with sub- and super-solutions does not work directly and we follow the method developed for proving [21, Theorem 1.9] which is a delicate extension of the one in the subcritical case. The fact that a sequence of approximation {un} is bounded in W01,q⁢(Ω)∩Lp⁢(Ω) does not imply the uniform integrability of {∇⁡un} in Lq⁢(Ω) for q≥NN-1.

Definition 3.3.

If r>1 and k∈ℕ*, we denote by 𝐌k,r⁢(Ω) the set of bounded measures μ in Ω which satisfy, for some C>0,

(3.15) | μ | ( E ) | ≤ C cap k , r Ω ( E ) for all compact sets E ⊂ Ω .

If Ω is replaced by ℝN, the set is denoted by 𝐌k,r. The smallest constant C such that (3.15) holds is denoted by [μ]Mk,r.

For T>1 we denote by E1⁢(T,μ) the subset of functions ζ∈W01,q⁢(Ω) such that

∫ Ω | ∇ ⁡ ζ | q ⁢ w ⁢ 𝑑 x ≤ T ⁢ ϵ q ⁢ ∫ Ω ( 𝐈 1 2 ⁢ R ⁢ [ | μ | ] ) q ⁢ w ⁢ 𝑑 x for all ⁢ w ∈ 𝐀 1 ∩ L ∞ .

The following estimate is obtained in [21, Lemma 5.2] in a more general context.

Lemma 3.4.

Let q>1 and μ∈M1,q′⁢(Ω). Then there exists c17=c17⁢(N,q)>0 such that for any ζ∈E1⁢(T,μ),

(3.16) 𝐈 1 2 ⁢ R ⁢ [ | ∇ ⁡ ζ | q ⁢ χ Ω ] ⁢ ( x ) ≤ c 17 ⁢ T ⁢ ϵ q ⁢ [ μ ] M 1 , q ′ q - 1 ⁢ 𝐈 1 2 ⁢ R ⁢ [ | μ | ] ⁢ ( x )

a.e. in Ω.

Lemma 3.5.

Let q>1, let ζ∈E1⁢(T,μ), where μ∈M1,q′⁢(Ω), and let S1⁢(ζ) be the solution of

- Δ ⁢ ϕ + ϕ p = M ⁢ | ∇ ⁡ ζ | q + ϵ ⁢ μ in ⁢ Ω ,

ϕ = 0 on ⁢ ∂ ⁡ Ω .

Then there exists c18=c18⁢(N,q)>0 such that

v ≤ S 1 ⁢ ( ζ ) ≤ S ⁢ ( ζ ) ≤ ( ϵ + c 18 ⁢ ϵ q ⁢ T ) ⁢ I 2 ⁢ [ μ ] ,

where S⁢(ζ) is the solution of

- Δ ⁢ ϕ = M ⁢ | ∇ ⁡ v | q + ϵ ⁢ μ in ⁢ Ω ,

ϕ = 0 on ⁢ ∂ ⁡ Ω

and v is the solution of (3.1) with μ replaced by ϵ⁢μ.

Proof.

There holds

S 1 ⁢ ( ζ ) = ϵ ⁢ 𝐆 Ω ⁢ [ μ ] + 𝐆 Ω ⁢ [ | ∇ ⁡ ζ | q ] ≤ ϵ ⁢ 𝐈 2 ⁢ [ μ ] + 𝐈 2 ⁢ [ | ∇ ⁡ ζ | q ] = ϵ ⁢ 𝐈 2 ⁢ [ μ ] + 𝐈 1 ⁢ [ 𝐈 1 ⁢ [ | ∇ ⁡ ζ | q ] ] .

By Lemma 3.4 with R=∞, we have 𝐈1⁢[|∇⁡ζ|q⁢χΩ]≤c17⁢T⁢ϵq⁢[μ]M1,q′q-1⁢𝐈1⁢[μ], hence

S 1 ⁢ ( ζ ) ≤ ( ϵ + c 17 ⁢ T ⁢ ϵ q ⁢ [ μ ] M 1 , q ′ q - 1 ) ⁢ 𝐈 2 ⁢ [ μ ] := ( ϵ + c 18 ⁢ ϵ q ⁢ T ) ⁢ 𝐈 2 ⁢ [ μ ] ,

since T≥1 and ϵ∈(0,1]. ∎

Lemma 3.6.

There exists T1>0 such that for any T>T1, there exists ϵT>0 such that for all ϵ∈(0,ϵT], S1 maps E⁢(T,μ) into itself.

Proof.

By [21, Theorem 1.4], and using the proof of [21, Theorem 1.9], there exist constants c19,c20,c21>0, depending on N, q, R and m, such that for any w∈𝐀1∩L∞ such that [w]A1≤m

(3.17)

∫ Ω | ∇ ⁡ S 1 ⁢ ( ζ ) | q ⁢ w ⁢ 𝑑 x ≤ c 19 ⁢ ∫ Ω ( 𝐈 1 2 ⁢ R ⁢ [ | ∇ ⁡ ζ | q ⁢ χ Ω ] + ϵ ⁢ 𝐈 1 2 ⁢ R ⁢ [ μ ] + 𝐈 1 2 ⁢ R ⁢ [ S 1 p ⁢ ( ζ ) ] ) q ⁢ w ⁢ 𝑑 x

≤ c 20 ⁢ ∫ Ω ( ( 𝐈 1 2 ⁢ R ⁢ [ | ∇ ⁡ ζ | q ⁢ χ Ω ] ) q + ϵ q ⁢ ( 𝐈 1 2 ⁢ R ⁢ [ μ ] ) q + ( ϵ p ⁢ q + c 21 ⁢ ϵ p ⁢ q 2 ⁢ T p ⁢ q ) ⁢ ( 𝐈 1 2 ⁢ R ⁢ [ ( 𝐈 2 ⁢ [ μ ] ) p ] ) q ) ⁢ w ⁢ 𝑑 x .

Using again the equivalence between (3.11), (3.12), (3.13) and (3.14), inequality (3.17) is transformed to

∫ Ω | ∇ ⁡ S 1 ⁢ ( ζ ) | q ⁢ w ⁢ 𝑑 x ≤ c 22 ⁢ ( ϵ q + c 24 ⁢ q ⁢ T q ⁢ ϵ q 2 + c 25 q ⁢ ( ϵ p ⁢ q + c 20 ⁢ ϵ p ⁢ q 2 ⁢ T p ⁢ q ) ) ⁢ ∫ Ω ( 𝐈 1 2 ⁢ R ⁢ [ μ ] ) q ⁢ w ⁢ 𝑑 x .

Finally, for any T>c23 there exists ϵ0 such that for 0<ϵ≤ϵ0, there exist positive constants c24 and c25 such that

c 22 ⁢ ( ϵ q + c 24 ⁢ q ⁢ T q ⁢ ϵ q 2 + c 25 q ⁢ ( ϵ p ⁢ q + c 20 ⁢ ϵ p ⁢ q 2 ⁢ T p ⁢ q ) ) ≤ T ⁢ ϵ q ,

which implies the claim. ∎

Lemma 3.7.

Under the assumptions of Lemma 3.6 with T>c23 and 0<ϵ≤ϵT, the mapping S1 is compact from E⁢(T,μ) into itself.

Proof.

We prove first the continuity. Let {ζj}⊂E⁢(T,μ) such that ζj→ζ in W01,q⁢(Ω). Using monotonicity as in [12, Theorem 8], we obtain that for any r∈[1,NN-1), there exists α=α⁢(N,R,r)>0 such that

(3.18) α ⁢ ∥ S 1 ⁢ ( ζ j ) - S 1 ⁢ ( ζ ℓ ) ∥ W 0 1 , r + ∥ S 1 p ⁢ ( ζ j ) - S 1 p ⁢ ( ζ ℓ ) ∥ L 1 ≤ M ⁢ ∥ | ∇ ⁡ ζ j | q - | ∇ ⁡ ζ ℓ | q ∥ L 1 .

Since {∇⁡ζj} is a Cauchy sequence in Lq⁢(Ω), it follows that S1⁢(ζj)→S1⁢(ζ) in W01,r⁢(Ω)∩Lp⁢(Ω). Hence

- Δ ⁢ S 1 ⁢ ( ζ ) + S 1 p ⁢ ( ζ ) = M ⁢ | ∇ ⁡ ζ | q + μ in ⁢ Ω ,

S 1 ⁢ ( ζ ) = 0 in ⁢ ∂ ⁡ Ω .

If we take ν:=νj=M⁢|∇⁡ζj|q+μ+S1p⁢(ζj), then by (3.13),

𝐈 1 2 ⁢ R ⁢ [ S 1 p ⁢ ( ζ j ) ] ≤ 𝐈 1 2 ⁢ R ⁢ [ ( 𝐈 2 ⁢ [ μ ] ) p ] ≤ C 6 ⁢ 𝐈 1 2 ⁢ R ⁢ [ μ ] .

Combined with (3.16) we infer that

𝐈 1 2 ⁢ R ⁢ [ M ⁢ | ∇ ⁡ ζ j | q + S 1 p ⁢ ( ζ j ) + μ ] ≤ c 28 ⁢ 𝐈 1 2 ⁢ R ⁢ [ μ ] .

Let 𝐌1 be the first order fractional maximal function defined by

𝐌 1 ⁢ ( μ ) ⁢ ( x ) := sup ρ > 0 ⁡ ω ⁢ ( B ρ ⁢ ( x ) ) ρ N - 1 for all ⁢ x ∈ ℝ N .

It is classical that for all ν∈𝔐b⁢(Ω),

𝐌 1 [ | ν | ] ( x ) ≤ m N 𝐈 1 2 ⁢ R [ | ν | ] ) ( x ) for a.e. x ∈ ℝ N .

Hence, if we set c29=mN⁢(c28+1), there holds

(3.19) 𝐌 1 ⁢ [ M ⁢ | ∇ ⁡ ζ j | q + μ + S 1 p ⁢ ( ζ j ) ] ⁢ ( x ) ≤ c 29 ⁢ 𝐈 1 2 ⁢ R ⁢ [ μ ] .

Since 𝐈12⁢R⁢[μ]∈Lq⁢(Ω), we deduce that the left-hand side of (3.19) is bounded and equi-integrable in Lq⁢(Ω). Then we apply [21, Corollary 1.7] with w=1 and deduce that there exists a subsequence {∇⁡S1⁢(ζjn)} which converges in Lq⁢(Ω). By uniqueness the limit is {∇⁡S1⁢(ζ)} and the whole sequence {∇⁡S1⁢(ζj)} converges. Therefore S1 is continuous.

The proof of the compactness follows the same ideas: If {ζj} is a bounded sequence in E⁢(T,μ), then {∇⁡S1⁢(ζj)} is bounded in Lwq⁢(Ω) by (3.17), hence (3.18) holds. By (3.12), {S1⁢(ζj)} is bounded in Lp⁢(Ω) and equi-integrable since 𝐈2⁢[μ] belongs to Lp⁢(Ω) (it is a consequence of (3.14)) and Lemma 3.2. By [11] the sequence {S1⁢(ζj)} is relatively compact in W01,1⁢(Ω). Hence, there exist ζ∈E⁢(T,μ) and a subsequence {ζjn} such that ζjn→ζ weakly in W01,q⁢(Ω) and {S1⁢(ζjn)} converges to some S∈E⁢(T,μ) in W01,1⁢(Ω)∩Lp⁢(Ω), a.e. in Ω and weakly in W01,q⁢(Ω). As above we derive from [21, Corollary 1.7] that the sequence {𝐌1⁢[M⁢|∇⁡ζjn|q+μ+S1p⁢(ζj)]} is bounded and equi-integrable in Lq⁢(Ω), hence {∇⁡S1⁢(ζjn)} converges to ∇⁡S in Lq⁢(Ω). Therefore S is a solution of

- Δ ⁢ S + S p = M ⁢ | ∇ ⁡ ζ | q + μ in ⁢ Ω ,

S = 0 on ⁢ ∂ ⁡ Ω .

This implies that S=S1⁢(ζ) and the mapping S1 is compact.∎

End of the proof of Theorem 1.3.

It follows from Lemma 3.7 that S1 is a compact mapping from E⁢(T,μ) into itself. Hence it admits a fixed point u by Schauder’s theorem and u∈W01,q⁢(Ω)∩Lp⁢(Ω) is nonnegative and satisfies (1.7).∎

3.3 Proof of the Corollaries

Proof of Corollary 1.4.

If q satisfies N⁢pN+p≤q<2, then 1<p<2⁢NN-2. By the Sobolev imbedding theorem there holds

∥ ϕ ∥ W 1 , q ′ ≤ c 27 ⁢ ∥ ϕ ∥ W 2 , p ′ for all ⁢ ϕ ∈ C 2 ⁢ ( Ω ¯ ) ,

where c27 depends on p, q and |Ω|, provided

1 q ′ ≥ 1 p ′ - 1 N ⇔ q ≥ N ⁢ p N + p .

The condition q<2 necessitates that N⁢pN+p<2, equivalently p<2⁢NN+2. ∎

Proof of Corollary 1.5.

We recall [1, Theorem 5.5.1 (a)–(b)]. If 2⁢p′≤q′<N, then

( cap 2 , p ′ ℝ N ⁡ ( E ) ) 1 N - 2 ⁢ p ′ ≤ A ⁢ ( cap 1 , q ′ ℝ N ⁡ ( E ) ) 1 N - q ′ for all Borel sets ⁢ E ⊂ Ω .

Since N-2⁢p′N-q′≥1, we deduce

cap 2 , p ′ ℝ N ⁡ ( E ) ≤ A ′ ⁢ cap 1 , q ′ ℝ N ⁡ ( E ) for all Borel sets ⁢ E ⊂ Ω .

Hence (1.9) implies (1.6) and existence follows from Theorem 1.3. The assumption 2⁢p′≤q′<N is equivalent to NN-1<q≤2⁢pp+1. Note that this implies p>NN-2. ∎

Proof of Corollary 1.6.

(i) Since p and q are subcritical, formula (1.6) is verified as soon as there holds for every Borel set E⊂Ω,

μ ⁢ ( E ) ≤ μ ⁢ ( Ω ) min ⁡ { cap 2 , p ′ ℝ N ⁡ ( { 0 } ) , cap 1 , q ′ ℝ N ⁡ ( { 0 } ) } ⁢ min ⁡ { cap 2 , p ′ ℝ N ⁡ ( E ) , cap 1 , q ′ ℝ N ⁡ ( E ) } .

(ii) If p is subcritical and q supercritical, then (1.8) implies (1.6).

(iii) If q is subcritical and μ satisfies (1.9), then inequality (1.6) holds and the existence follows by Theorem 1.3. However, the condition that μ vanishes on Borel sets with cap2,p′ℝN-capacity is weaker. Let T>0 and let BT be the set of ζ∈W01,q⁢(Ω) such that ∥M⁢|∇⁡ζ|q∥L1⁢(Ω)≤T. If ζ∈BT, then there exists a unique solution S⁢(ζ)∈W01,1⁢(Ω)∩Lp⁢(Ω) to

- Δ ⁢ ϕ + ϕ p = M ⁢ | ∇ ⁡ ζ | q + μ in ⁢ Ω ,

ϕ = 0 on ⁢ ∂ ⁡ Ω .

This follows from [4] since the right-hand side of the equation is a measure absolutely continuous with respect to the cap2,p′ℝN-capacity and the solution is nonnegative. The following estimates hold [12, Theorem 8]:

(3.20)

∫ Ω S p ⁢ ( ζ ) ⁢ 𝑑 x ≤ μ ⁢ ( Ω ) + M ⁢ ∫ Ω | ∇ ⁡ ζ | q ⁢ 𝑑 x ,

α ⁢ ∥ S ⁢ ( ζ ) ∥ W 0 1 , r ≤ 2 ⁢ μ ⁢ ( Ω ) + 2 ⁢ M ⁢ ∫ Ω | ∇ ⁡ ζ | q ⁢ 𝑑 x ,

for any 1<r<NN-1. This implies in the case q=r that

( ∫ Ω | ∇ ⁡ S ⁢ ( ζ ) | q ⁢ 𝑑 x ) 1 q ≤ A ⁢ μ ⁢ ( Ω ) + B ⁢ ∫ Ω | ∇ ⁡ ζ | q ⁢ 𝑑 x ,

with

A = 2 α and B = 2 ⁢ M α .

For X>0 set F⁢(X)=B⁢Xq-X+A⁢μ⁢(Ω). Then F′⁢(X)=0 if and only if X=X0:=(q⁢B)-1q-1 and

F ⁢ ( X 0 ) = A ⁢ μ ⁢ ( Ω ) - q - 1 q ⁢ ( q ⁢ B ) - 1 q - 1 .

If we assume that

F ⁢ ( X 0 ) < 0 ⇔ μ ⁢ ( Ω ) < q - 1 q ⁢ A ⁢ ( q ⁢ B ) - 1 q - 1 = ( q - 1 ) ⁢ ( α 2 ) q q - 1 ⁢ M - 1 q - 1 ,

then min⁡F⁢(X)<0, therefore F admits two positive roots X2<X0<X1. Hence the inequality

X 2 < ( ∫ Ω | ∇ ⁡ ζ | q ⁢ 𝑑 x ) 1 q ≤ X 1

implies

( ∫ Ω | ∇ ⁡ S ⁢ ( ζ ) | q ⁢ 𝑑 x ) 1 q ≤ B ⁢ ∫ Ω | ∇ ⁡ ζ | q ⁢ 𝑑 x + A ⁢ μ ⁢ ( Ω ) ≤ ( ∫ Ω | ∇ ⁡ ζ | q ⁢ 𝑑 x ) 1 q ≤ X 1 ,

and if

( ∫ Ω | ∇ ⁡ ζ | q ⁢ 𝑑 x ) 1 q ≤ X 2 ,

then

( ∫ Ω | ∇ ⁡ S ⁢ ( ζ ) | q ⁢ 𝑑 x ) 1 q ≤ A ⁢ μ ⁢ ( Ω ) + B ⁢ X 2 q ≤ A ⁢ μ ⁢ ( Ω ) + B ⁢ X 1 q = X 1 .

Therefore the mapping S sends BX1 into itself. It is continuous since

∫ Ω | S p ⁢ ( ζ 1 ) - S p ⁢ ( ζ 2 ) | ≤ M ⁢ ∫ Ω | | ∇ ⁡ ζ 1 | q - | ∇ ⁡ ζ 2 | q | ⁢ 𝑑 x

α ⁢ ∥ S ⁢ ( ζ 1 ) - S ⁢ ( ζ 2 ) ∥ W 0 1 , r ≤ 2 ⁢ M ⁢ ∫ Ω | | ∇ ⁡ ζ 1 | q - | ∇ ⁡ ζ 2 | q | ⁢ 𝑑 x .

The fact that this operator is compact follows from the a priori estimate (3.20) and Lemma 3.7. The conclusion is a consequence of the Schauder Theorem. ∎

Communicated by Shair Ahmad

Funding statement: This article has been prepared with the support of the FONDECYT grants 1210241 and 1190102 for the three authors.

References

[1] D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Grundlehren Math. Wiss. 314, Springer, Berlin, 1996. 10.1007/978-3-662-03282-4Search in Google Scholar

[2] S. Alarcón, J. García-Melián and A. Quaas, Keller–Osserman type conditions for some elliptic problems with gradient terms, J. Differential Equations 252 (2012), no. 2, 886–914. 10.1016/j.jde.2011.09.033Search in Google Scholar

[3] C. Bandle and E. Giarrusso, Boundary blow up for semilinear elliptic equations with nonlinear gradient terms, Adv. Differential Equations 1 (1996), no. 1, 133–150. 10.57262/ade/1366896318Search in Google Scholar

[4] P. Baras and M. Pierre, Singularités éliminables pour des équations semi-linéaires, Ann. Inst. Fourier (Grenoble) 34 (1984), no. 1, 185–206. 10.5802/aif.956Search in Google Scholar

[5] M.-F. Bidaut-Véron, Liouville results and asymptotics of solutions of a quasilinear elliptic equation with supercritical source gradient term, Adv. Nonlinear Stud. 21 (2021), 57–76. 10.1515/ans-2020-2109Search in Google Scholar

[6] M.-F. Bidaut-Véron, M. García-Huidobro and L. Véron, Estimates of solutions of elliptic equations with a source reaction term involving the product of the function and its gradient, Duke Math. J. 168 (2019), no. 8, 1487–1537. 10.1215/00127094-2018-0067Search in Google Scholar

[7] M.-F. Bidaut-Véron, M. Garcia-Huidobro and L. Véron, Boundary singular solutions of a class of equations with mixed absorption-reaction, preprint (2020), https://arxiv.org/abs/2007.16097v2. 10.1007/s00526-022-02200-zSearch in Google Scholar

[8] M.-F. Bidaut-Véron, M. Garcia-Huidobro and L. Véron, Singular solutions of some elliptic equations involving mixed absorption-reaction, in preparation. 10.3934/dcds.2022036Search in Google Scholar

[9] M.-F. Bidaut-Véron, Q.-H. Nguyen and L. Véron, Quasilinear elliptic equations with a source reaction term involving the function and its gradient and measure data, Calc. Var. Partial Differential Equations 59 (2020), no. 5, Paper No. 148. 10.1007/s00526-020-01808-3Search in Google Scholar

[10] L. Boccardo, F. Murat and J.-P. Puel, Résultats d’existence pour certains problèmes elliptiques quasilinéaires, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 11 (1984), no. 2, 213–235. Search in Google Scholar

[11] H. Brezis, Analyse fonctionnelle et Applications, Masson, Paris, 1983. Search in Google Scholar

[12] H. Brézis and W. A. Strauss, Semi-linear second-order elliptic equations in L1, J. Math. Soc. Japan 25 (1973), 565–590. 10.2969/jmsj/02540565Search in Google Scholar

[13] H. Brézis and L. Véron, Removable singularities for some nonlinear elliptic equations, Arch. Ration. Mech. Anal. 75 (1980/81), no. 1, 1–6. 10.1007/BF00284616Search in Google Scholar

[14] P. Felmer, A. Quaas and B. Sirakov, Solvability of nonlinear elliptic equations with gradient terms, J. Differential Equations 254 (2013), no. 11, 4327–4346. 10.1016/j.jde.2013.03.003Search in Google Scholar

[15] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Class. Math., Springer, Berlin, 2001. 10.1007/978-3-642-61798-0Search in Google Scholar

[16] K. Hansson, V. G. Maz’ya and I. E. Verbitsky, Criteria of solvability for multidimensional Riccati equations, Ark. Mat. 37 (1999), no. 1, 87–120. 10.1007/BF02384829Search in Google Scholar

[17] J. B. Keller, On solutions of Δ⁢u=f⁢(u), Comm. Pure Appl. Math. 10 (1957), 503–510. 10.1002/cpa.3160100402Search in Google Scholar

[18] P.-L. Lions, Quelques remarques sur les problèmes elliptiques quasilinéaires du second ordre, J. Anal. Math. 45 (1985), 234–254. 10.1007/BF02792551Search in Google Scholar

[19] V. G. Maz’ya and I. E. Verbitsky, Capacitary inequalities for fractional integrals, with applications to partial differential equations and Sobolev multipliers, Ark. Mat. 33 (1995), no. 1, 81–115. 10.1007/BF02559606Search in Google Scholar

[20] P.-T. Nguyen, Isolated singularities of positive solutions of elliptic equations with weighted gradient term, Anal. PDE 9 (2016), no. 7, 1671–1692. 10.2140/apde.2016.9.1671Search in Google Scholar

[21] Q.-H. Nguyen and N. C. Phuc, Good-λ and Muckenhoupt–Wheeden type bounds in quasilinear measure datum problems, with applications, Math. Ann. 374 (2019), no. 1–2, 67–98. 10.1007/s00208-018-1744-2Search in Google Scholar

[22] R. Osserman, On the inequality Δ⁢u≥f⁢(u), Pacific J. Math. 7 (1957), 1641–1647. 10.2140/pjm.1957.7.1641Search in Google Scholar

[23] N. C. Phuc, Nonlinear Muckenhoupt–Wheeden type bounds on Reifenberg flat domains, with applications to quasilinear Riccati type equations, Adv. Math. 250 (2014), 387–419. 10.1016/j.aim.2013.09.022Search in Google Scholar

[24] N. C. Phuc and I. E. Verbitsky, Quasilinear and Hessian equations of Lane–Emden type, Ann. of Math. (2) 168 (2008), no. 3, 859–914. 10.4007/annals.2008.168.859Search in Google Scholar

[25] J. Serrin, Isolated singularities of solutions of quasi-linear equations, Acta Math. 113 (1965), 219–240. 10.1007/BF02391778Search in Google Scholar

[26] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser. 30, Princeton University, Princeton, 1970. 10.1515/9781400883882Search in Google Scholar

[27] L. Véron, Singular solutions of some nonlinear elliptic equations, Nonlinear Anal. 5 (1981), no. 3, 225–242. 10.1016/0362-546X(81)90028-6Search in Google Scholar

Received: 2021-02-26

Accepted: 2021-03-04

Published Online: 2021-03-18

Published in Print: 2021-05-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Measure Data Problems for a Class of Elliptic Equations with Mixed Absorption-Reaction

Abstract

1 Introduction

Theorem 1.1.

Theorem 1.2.

Theorem 1.3.

Corollary 1.4.

Corollary 1.5.

Corollary 1.6.

2 Removable Singularities

2.1 A Priori Estimates

Proposition 2.1.

Proof.

Corollary 2.2.

Proposition 2.3.

Proof.

Remark.

Lemma 2.4.

Proof.

Remark.

Corollary 2.5.

Proposition 2.6.

Proof.

Proposition 2.7.

Proof.

Remark.

Proposition 2.8.

Remark.

2.2 Proof of Theorem 1.1

Step 1.

Step 2.

Step 3.

Proposition 2.9.

Remark.

2.3 Proof of Theorem 1.2

3 Measure Data

3.1 Proof of Theorem 1.3: The Case 1<q<NN-1

Theorem 3.1.

Lemma 3.2.

Proof.

Proof of Theorem 3.1.

3.2 Proof of Theorem 1.3: The General Case

Definition 3.3.

Lemma 3.4.

Lemma 3.5.

Proof.

Lemma 3.6.

Proof.

Lemma 3.7.

Proof.

End of the proof of Theorem 1.3.

3.3 Proof of the Corollaries

Proof of Corollary 1.4.

Proof of Corollary 1.5.

Proof of Corollary 1.6.

References

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