An existence result for anisotropic quasilinear problems

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Abstract

We study existence of solutions for a boundary degenerate (or singular) quasilinear equation in a smooth bounded domain under Dirichlet boundary conditions. We consider a weighted p-Laplacian operator with a coefficient that is locally bounded inside the domain and satisfying certain additional integrability assumptions. Our main result applies for boundary value problems involving continuous non-linearities having no growth restriction, but provided the existence of a sub and a supersolution is guaranteed. As an application, we present an existence result for a boundary value problem with a non-linearity f(u) satisfying f(0)0 and having (p1)-sublinear growth at infinity.

Introduction

In this work we study the degenerate (or singular) boundary value problem (BVP) diva(x)|u|p2u=F(x,u)inΩu=0onΩ,where N2, p(1,N) and ΩRN is a smooth bounded domain.

In order to formulate our assumptions on the function a:ΩR, we consider first ρ0>0 small so that in the set Ωρ0{xΩ:0<dist(x,Ω)<ρ0} the distance function Ωρ0xdist(x,Ω)satisfies that given any xΩρ0, there exists a unique zxΩ such that dist(x,Ω)=dist(x,zx) (see also Section 2.3). We consider also a positive function a:(0,ρ0)(0,) and a fixed positive number s such that aL1(0,ρ0)Lloc(0,ρ0),a1Ls(0,ρ0)ands>maxNp,1p1.

In particular, a1p1L1(0,ρ0).

The assumptions for the function a are the following:

(A1) aLloc(Ω) and for any open set D with D¯Ω, infDa>0;

(A2) for every xΩρ0, a(x)=a(dist(x,Ω)).

Assumptions (A1) and (A2) imply that the differential operator in (1.1) is uniformly elliptic on any subdomain of Ω, but not necessarily near the boundary of the domain.

These hypotheses also yield that aL1(Ω) and a1Ls(Ω) (see Section 2.1), which will allow us to put Eq. (1.1) into an appropriate functional analytic setting.

To formulate the assumptions on the function F:Ω×RR, we first let s>0 be as in (1.3) and denote pspss+1,psNpsNps=NpsN(s+1)ps.

Using that s>Np>1, we find that ps<p<ps<NpNp.

Next, fix q[p,ps) and set qqp, if q=p. Consider also a measurable function b satisfying the following:

(B1) bLqqp(Ω);

(B2) there exist constants 0<c1<c2 such that c1lim infdist(x,Ω)0a1p1(x)b(x)lim supdist(x,Ω)0a1p1(x)b(x)c2.

In particular, b0 and b0 a.e. in Ω.

As for the function F:Ω×RR, we assume it is a Caratheodory function satisfying the growth condition:

for every M>0, there exists CM>0 such that for every ζR with |ζ|M and for a.e. xΩ, |F(x,ζ)|CMb(x).

We use weighted Sobolev spaces to study the BVP (1.1). The Sobolev space W1,p(Ω,a) is defined as the class of functions vWloc1,p(Ω) such that vW1,p(Ω,a)Ω|v|pdx+Ωa(x)|v|pdx1p<.

The Sobolev space W01,p(Ω,a) is defined as the closure of Cc(Ω) with respect to the norm W1,p(Ω,a).

A function uW01,p(Ω,a) is called a solution of (1.1) if F(,u())Lpp1(Ω) and φW01,p(Ω,a):Ωa(x)|u|p2uφdx=ΩF(x,u)φdx.

A function uW1,p(Ω,a)C(Ω¯) is called a subsolution (supersolution respectively) of (1.1) if F(,u())Lpp1(Ω), for every φW01,p(Ω,a) with φ0 a.e. in Ω, Ωa(x)|u|p2uφdx()ΩF(x,u)φdxand maxxΩu(x)0  minxΩu(x)0.

Our first main result reads as follows.

Theorem 1.1

Let (A1)-(A2) and (B1)-(B2) hold and let F satisfy (1.6). Assume that u̲ is a subsolution and U¯ is a supersolution to (1.1) such that u̲U¯ in Ω. Then the BVP (1.1) has a solution uC(Ω¯) such that u̲uU¯ in Ω.

The proof of Theorem 1.1 is motivated by the techniques described in Chapter 5 in [1]. As an application of Theorem 1.1, we present existence of a positive solution for the BVP diva(x)|u|p2u=λb(x)f(u)inΩu=0onΩ,where λ>0 is a positive parameter and f:[0,)R is a continuous function satisfying the following hypotheses:

(f1) f is (p1)-sublinear, i.e., lim supζf(ζ)ζp1=0;

(f2) there exists r(0,p1) such that lim infζf(ζ)ζr(0,].

(f3) f(0)0 and min0ζ<f(ζ)<0.

Problem (1.9) finds its applications, for instance, in resource management models, see [2], and design of suspension bridges, see [3].

We present the function f(ζ)=A+ζp1log(2+ζ)forζ0,where A0, as an example of a nonlinearity satisfying conditions (f1), (f2) and (f3).

Condition f(0)0 in (f3) implies that u=0 is a supersolution to (1.9). Under this condition, finding a positive subsolution for (1.9) becomes a subtle matter. We refer the reader to [4], [5], [6], [7], [8] and references therein, for results concerned with related boundary value problems in the uniformly elliptic case.

The authors in [9] showed the existence of a positive solution of Eq. (1.9) either in a ball or in RN, under a similar set of assumptions on a,b and f, but assuming in addition the global radial symmetry of the coefficient a and instead of (f3), assuming that there exist K0,δ>0 and γ(0,1) such that for everyζ(0,δ),K0ζγf(ζ)0.

As remarked in [9], our hypotheses do not allow, in general, to obtain either Cloc1-regularity of weak solutions to (1.9), see [10], or Hopf’s boundary lemma, see [11]. Instead, regularity of solutions relies on the local Hölder continuity due to Serrin, see [12]. In particular, global a priori C1-estimates and therefore the construction of a subsolution of (1.9) is a much more challenging task. In this work, tools from differential geometry help us overcome this difficulty.

More precisely, we say that Ω has non-negative mean curvature if for every xΩ, i=1N1ki(x)0,where k1,,kN1 are the principal curvatures of Ω.

This geometric notion generalizes convexity and it is equivalent to the monotonicity of the surface area element of Ω, see [13]. This monotonicity states that the (N1)-dimensional volume of every subdomain ωΩ does not decrease as ω approaches Ω.

From the PDE point of view, condition (1.10) implies that the distance function dist(,Ω) is subharmonic in any subdomain of Ωρ0 (see (1.2)) and in essence this fact will allow us to construct an appropriate subsolution to (1.9).

Our second main result reads as follows.

Theorem 1.2

Let Ω be a smooth bounded domain such that Ω has non-negative mean curvature. Let (A1)-(A2) and (B1)-(B2) hold and let fC[0,) satisfy (f1)-(f3). Then, there exists λ0>0 such that for every λλ0 the BVP (1.9) has a solution uC(Ω¯) which is positive in Ω.

We refer the reader to [14], [15] for similar results in the uniformly elliptic setting.

In the case a(x)1 and p=2, under slightly more restrictive assumptions on f and with no assumptions on the geometry of Ω, the authors in [16] proved uniqueness of positive solutions of (1.9) for λ>0 large. We also refer the reader to [17] for non-existence of positive radial solutions of a related BVP in the superlinear setting.

We have considered assumption (A2) for the sake of clarity in our presentation. Although, we remark that Theorem 1.1, Theorem 1.2 are still valid if instead of (A2), we assume the existence of functions a1,a2:(0,ρ0)(0,), both satisfying (1.3) and such that a1(dist(x,Ω))a(x)a2(dist(x,Ω))for every xΩρ0. The proofs require only straight forward changes and details are left to the reader.

The paper is organized as follows. In Section 2 we present the preliminary results and a priori estimates needed for the proof of Theorem 1.1. The proof of Theorem 1.1 is presented in Section 3. In Section 4 we provide the proof Theorem 1.2.

Section snippets

Weighted Sobolev spaces

Let p(1,N), ΩRN be a smooth bounded domain and let a:ΩR satisfy (A1) and (A2).

From these hypotheses and with the help of Fermi coordinates, we verify in Section 2.3 that aL1(Ω) and a1Ls(Ω). Observe also that the Hölder inequality and the fact that s>1p1 yield a1p1L1(Ω).

Thus, the spaces W1,p(Ω,a) and W01,p(Ω,a), defined in Section 1, are separable. They are also uniformly convex and therefore reflexive Banach spaces (see Chapter 1 in [18]).

We also notice from (A1) that for any D with D¯

Proof of Theorem 1.1

In this part we follow the scheme from Chapter 5 in [1] to prove Theorem 1.1. Also, we make use of the notations and conventions introduced in Sections 1 Introduction, 2 Preliminaries.

Assume hypotheses (A1)–(A2) and (B1)–(B2). Using (B2), let ρ1(0,ρ0) be such that xΩρ1:c1a1p1(x)b(x)c2a1p1(x),where Ωρ1 is as in (2.24).

Consider the space C(Ω¯) endowed with the norm L(Ω) and its subspace C0(Ω¯){wC(Ω¯):w=0onΩ}.

Next, consider the space Yab̃Lqqp(Ω):a1p1|b̃|L(Ωρ1)where q[p,ps)

Proof of Theorem 1.2

In this final section we prove Theorem 1.2. Besides (A1)-(A2) and (B1)-(B2), we assume also (f1)-(f3). We adopt the notations and conventions from the previous sections.

Proposition 4.1

For every λ>0, there exist U¯λW1,p(Ω,a)C0(Ω¯) a supersolution to (1.9) with U¯λ>0 in Ω.

Although the proof of Proposition 4.1 is rather standard, we include it here for the sake of completeness.

Proof

Let λ>0 be arbitrary. Set U¯=Mψ, where ψ is the solution to (3.2) with b̃=b and M>0 is a parameter to be chosen later. Since b(x)0,

Acknowledgment

The first author was supported by the Project LO1506 of the Ministry of Education, Youth and Sports of the Czech Republic .

The authors want to thank the referee for careful reading of the manuscript and the suggestions which improved the exposition of the text.

References (24)

  • BrownK.J. et al.

    Nonexistence of radially symmetric nonnegative solutions for a class of semi-positone problems

    Differential Integral Equations

    (1989)
  • VazquezJ.L.

    A strong maximum principle for some quasilinear elliptic equations

    Appl. Math. Optim.

    (1984)
  • Cited by (0)

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