Positive radial symmetric solutions for a class of elliptic problems with critical exponent and -1 growth

Chun-Yu Lei; Jia-Feng Liao

doi:10.1515/anona-2020-0174

Open Access Published by De Gruyter April 14, 2021

Positive radial symmetric solutions for a class of elliptic problems with critical exponent and -1 growth

Chun-Yu Lei and Jia-Feng Liao

From the journal Advances in Nonlinear Analysis

https://doi.org/10.1515/anona-2020-0174

Abstract

In this paper, we consider a class of semilinear elliptic equation with critical exponent and -1 growth. By using the critical point theory for nonsmooth functionals, two positive solutions are obtained. Moreover, the symmetry and monotonicity properties of the solutions are proved by the moving plane method. Our results improve the corresponding results in the literature.

Keywords: Singular elliptic equation; critical growth; radial symmetry; critical point theory for nonsmooth functionals; moving plane method

MSC 2010: 35B33; 35J61; 35J75

1 Introduction and main result

Consider the multiplicity of positive solutions for the following singular elliptic equation with critical growth

Δu+u2∗−1+μu=0,inB,u=0,on∂B, (1.1)

where B⊂ℝ^N (N=3) is the unit ball, μ is a positive constant. Problem (1.1) has a variational structure given by the functional

I(u)=12∫B|∇u|2dx−12∗∫B|u|2∗dx−μ∫Bln⁡|u|dx

for u∈H01(B) . It is well known that the singular term leads to the non-differentiability of I on H01(B) . In fact, since I (tu) → +∞ as t → 0⁺, I is not continuous at the point 0. Therefore, it is difficult to find out the local minimizer and the mountain pass type solutions of problem (1.1). In order to find firstly a local minimizer solution, we consider the following problem

Δu+μu=0, in B, u=0, on ∂B. (1.2)

According to Theorem 1 of [1], we know that problem (1.2) has a unique positive solution wμ∈C2+α(B)∩C(B¯)(0<α<1) with w_μ ≥ cϕ₁ (where ϕ₁ is an eigenfunction corresponding to the smallest eigenvalue λ₁ of the problem Δϕ+μϕ=0, ϕ∣_∂B=0). Moreover, [1] proved that the following inequality holds

∫B1ϕ1tdx<+∞, (1.3)

if and only if t ∈ (0, 1).

The following singular elliptic problem has been extensively considered

Δu+up+μuγ−0, in Ω,u−0, on ∂Ω, (1.4)

where Ω⊂ℝ^N(N=3) is a bounded domain with smooth boundary ∂Ω, 0 < p ≤ 2⋆−1 and γ>0. For examples, [2, 3, 4, 5, 6, 7, 8, 9, 10, 11] studied the case of 0<γ<1 for problem (1.4). Particularly, by the variational method and Nehari method, Sun, Wu and Long investigated the multiplicity of positive solutions for the singular elliptic problem for the first time in [8]. And, Yang [11] discussed problem (1.4) with p=2⋆−1 for the first time, and obtained two positive solutions by using the variational method and sub-supersolution method. For the case γ>1, problem (1.4) is considered by [1,12,13].

To our best knowledge, problem (1.4) with γ=1 is only investigated by [4], and two positive solutions are obtained when 1 < p<2⋆−1. A nature question is whether there exist positive solutions for problem (1.4) with γ=1 and p=2⋆−1. In the present note, we give a positive answer by the critical point theory for nonsmooth functionals, and obtain two positive solutions for problem (1.1). Moreover, based on the moving plane technique, we study the symmetry and monotonicity properties of positive solutions to problem (1.1).

In order to study problem (1.1), we define f:B × ℝ → [0, +∞) by

f(x,t)=1/t, if x∈B and t≥wμ(x),1/wμ, if x∈B and t≤wμ(x).

Consider the following auxiliary problem

Δu+u2∗−1+μf(x,u)=0, in B, u=0, on ∂B. (1.5)

Problem (1.5) has a variational structure given by the functional

J(u)=12∥u∥2−12∗∫B(u+)2∗dx−μ∫BF(x,u+)dx,

where F(x,u+)=∫0u+f(x,t)dt . Then the functional J is only continuous in H01(B) . In the following, we need find out critical points of J, and prove that they are weak solutions of problem (1.1).

Now our main result is as follows.

Theorem 1.1

There exists µ⋆>0 such that for 0<μ<µ⋆, problem (1.1) has two positive radially symmetric solutions. Moreover, the solutions are monotone decreasing about the origin.

Remark 1

To the best of our knowledge, problem (1.1) has not been studied up to now. Compared with [4], we generalize the corresponding result to the case of critical exponent.

2 Proof of Theorem 1.1

We divide two parts to prove Theorem 1.1. First, by using the critical point theory for nonsmooth functionals, we prove that problem (1.1) has at least two positive solutions. Then, we prove that the solution of problem (1.1) is radially symmetric and monotone decreasing about the origin by the moving plane method.

2.1 Existence of Two Positive Solutions

By the definition of F, the following statement is valid.

Lemma 2.1

Assume that {u_n} is bounded in H01(B) and u_n⇀ u in H01(B) , then

limn→∞∫BF(x,un)dx=∫BF(x,u)dx. (2.1)

Proof

When u < w_μ, one has f(x,u)=1wμ , so F(x,u)=∫0u+f(x,t)dt≤1 . When u > w_μ, one has f(x,u)=1u=1u+ , note that ln ∣x∣ ≤ ∣x∣. Then

F(x,u)=∫0wμ1wμdt+∫wμu+1tdt=1+ln⁡u++2ln⁡1wμ≤1+u++2wμ.

From the above information, one has

F(x,u)≤1+u++2wμ, for u∈H01(B). (2.2)

From [1], we have wμ≥cϕ1 . By (1.3), it holds that

∫B1wμdx≤c∫B1ϕ1dx<+∞. (2.3)

By the dominated convergence theorem, (2.1) holds. The proof is complete. □

We now recall some concepts adapted from critical point theory for nonsmooth functionals. Let (X, d) be a complete metric space, f:X→R be a continuous functional in X. Denote by ∣Df∣(u) the supremun of δ in [0, ∞) such that there exist r > 0, a neighborhood U of u ∈ X, and a continuous map σ: U × [0, r] satisfying

f(σ(v,t))≤f(v)−δt,(v,t)∈U×[0,r],d(σ(v,t),v)≤t,(v,t)∈U×[0,r]. (2.4)

The extended real number ∣Df∣(u) is called the weak slope of f at u, see [14,15].

A sequence {u_n} of X is called Palais-Smale sequence of the functional f, if ∣Df∣(u_n) → 0 as n → ∞ and f(u_n) is bounded. We say that u ∈ X is a critical point of f if ∣Df∣(u)=0. Since u → ∣Df∣(u) is lower semicontinuous, any accumulation point of a (PS) sequence is clearly a critical point of f.

Since looking for positive solutions of problem (1.1), we consider the functional J as defined on the closed positive cone P of H01(B)

P={u|u∈H01(B),u(x)≥0, a.e. x∈B}.

P is a complete metric space and J is a continuous functional on P. Then we have the following conclusion.

Lemma 2.2

Assume that u ∈ P and ∣DJ∣(u)<+∞, then for any v ∈ P there holds

μ∫Bf(x,u)(v−u)dx≤∫B∇u∇(v−u)dx−∫Bu2∗−1(v−u)dx+|DJ|(u)∥v−u∥. (2.5)

Proof

Similar to the proof of Lemma 3.1 in [15]. Let ∣DJ∣(u) < c, δ<12∥v−u∥ , v ∈ P and v≠u. Define the mapping σ: U × [0, δ] → P by

σ(z,t)=z+tv−z∥v−z∥,

where U is a neighborhood of u. Then ∣∣σ(z, t) − z∣∣=t, combining with (2.4), there exists a pair (z, t) ∈ U × [0, δ] such that

J(σ(z,t))>J(z)−ct.

Consequently, we assume that there exist sequences {u_n}⊂ P and {t_n}⊂[0, +∞), such that u_n → u, t_n → 0⁺, and

Jun+tnv−un∥v−un∥≥J(un)−ctn,

that is,

J(un+sn(v−un))≥J(un)−csn∥v−un∥, (2.6)

where sn=tn∥v−un∥→0+ as n → ∞. Divided by s_n in (2.6), one has

μ∫BF(x,un+sn(v−un))−F(x,un)sndx≤12∫B|∇(un+sn(v−un))|2−|∇un|2sndx−12∗∫B(un+sn(v−un))2∗−un2∗sndx+c∥v−un∥.

Set

I1,n=∫BF(x,un+sn(v−un))−F(x,(1−sn)un)sndx

and

I2,n=∫BF(x,(1−sn)un)−F(x,un)sndx.

Notice that

I1,n=∫Bf(x,ξn)snvsndx=∫Bf(x,ξn)vdx,

where ξ_n ∈ (u_n−s_nu_n, u_n+s_n(v − u_n)), which implies that ξ_n → u(u_n → u) as s_n → 0⁺. Note that F(x, t) is increasing in t, then I_1,n≥0 for all n. Applying Fatou’s Lemma to I_1,n, we obtain

lim infn→∞I1,n≥∫Bf(x,u)vdx

for v ∈ P. For I_2,n, by the differential mean value theorem, we have

lim n → ∞ I 2 , n = − ∫ B f ( x , u ) u d x .

From the above information, one has

μ∫Bf(x,u)(v−u)dx≤lim infn→∞(I1,n+I2,n)≤∫B∇u∇(v−u)dx−∫Bu2∗−1(v−u)dx+c∥v−u∥

for every v ∈ P. Since ∣DJ∣(u) < c is arbitrary, this leads us to the proof of Lemma 2.2. □

Lemma 2.3

(i)Assume that u is a critical point of J, then u is a weak solution of problem (1.5), that is, for all φ∈H01(B) it holds that

∫B∇u∇φdx=∫Bu2∗−1φdx+μ∫Bf(x,u)φdx. (2.7)

Moreover, u=w_μ a.e. in B.

(ii)If u is a critical point of J, then u is a positive solution of problem (1.1).

Proof

(i) Let u be a critical point of J. By Lemma 2.2, for φ∈H01(B) , s > 0, taking v=(u+sφ)⁺ ∈ P as test function in (2.5), one has

0≤∫B∇u∇((u+sφ)+−u)dx−∫Bu2∗−1((u+sφ)+−u)dx−μ∫Bf(x,u)((u+sφ)+−u)dx=s∫B∇u∇φdx−∫Bu2∗−1φdx−μ∫Bf(x,u)φdx−∫{u+sφ<0}∇u∇(u+sφ)dx+∫{u+sφ<0}u2∗−1(u+sφ)dx+μ∫{u+sφ<0}f(x,u)(u+sφ)dx≤s∫B∇u∇φdx−∫Bu2∗−1φdx−μ∫Bf(x,u)φdx−s∫{u+sφ<0}∇u∇φdx.

Since ∇ u(x)=0 for a.e. x ∈ B with u(x)=0 and meas{x ∈ B∣ u(x)+sφ(x) < 0, u(x) > 0} → 0 as s → 0, we have

∫{u+sφ<0}∇u∇φdx=∫{u+sφ<0,u>0}∇u∇φdx→0 as s→0.

Therefore

0≤s∫B∇u∇φdx−∫Bu2∗−1φdx−μ∫Bf(x,u)φdx+o(s)

as s → 0. We obtain

∫B∇u∇φdx−∫Bu2∗−1φdx−μ∫Bf(x,u)φdx≥0.

By the arbitrariness of the sign of φ, we can deduce that (2.7) holds. Thus, u is a weak solution of problem (1.5).

Next, we prove that u=w_μ a.e. in B. Choosing φ=(u − w_μ)⁻ in (2.7), one has

∫{u<wμ}(∇u,∇(u−wμ))dx=∫{u<wμ}(u2∗−1+μwμ)(u−wμ)dx. (2.8)

Notice that

∫{u<wμ}(∇wμ,∇(u−wμ))dx=μ∫{u<wμ}u−wμwμdx. (2.9)

Hence, it follows from (2.8) and (2.9) that

∫{u<wμ}|∇(u−wμ)|2dx=∫{u<wμ}u2∗−1(u−wμ)dx≤0,

which implies that ∣(u − w_μ)⁻∣=0, that is, u(x)=w_μ(x) a.e. in B. Thus, u ∈ P.

(ii) This follows from (i). The proof of Lemma 2.3 is completed. □

Let S be the best Sobolev constant, namely

S:=infu∈D1,2(RN)∖{0}∫RN|∇u|2dx∫RN|u|2∗dx22∗=infu∈H01(Ω)∖{0}∫Ω|∇u|2dx∫Ω|u|2∗dx22∗. (2.10)

Lemma 2.4

Assume that 0<μ<1 and J(un)→c<1NSN2−Θμ , ∣DJ∣(u_n) → 0 as n → ∞, then u_n → u in H01(B) , where Θ=Θ(N,S,|B|,∫B1ϕ1dx) .

Proof

Let {un}⊂P⊂H01(B) be such that ∣DJ∣(u_n) → 0, J(u_n) → c as n → ∞. By Lemma 2.2, we have

μ∫Bf(x,un)(v−un)dx≤∫B[∇un∇(v−un)−un2∗−1(v−un)]dx+|DJ|(un)∥v−un∥. (2.11)

Taking v=2u_n ∈ P in (2.11), we have

∥un∥2−∫Bun2∗dx−μ∫Bf(x,un)undx+|DJ|(un)∥un∥≥0. (2.12)

By (2.2) and (2.3)

J(un)+12∗|DJ|(un)∥un∥≥1N∥un∥2−μ∫BF(x,un)dx≥1N∥un∥2−c∥un∥,

for some positive constant c. Therefore, {u_n} is bounded in H01(Ω) . Thus, there exist a subsequence, still denoted by {u_n}, and u such that u_n⇀ u in H01(B) . Taking v=0 ∈ P in (2.11) again, one has

∥un∥2−∫Bun2∗dx−μ∫Bf(x,un)undx+|DJ|(un)∥un∥≤0. (2.13)

It follows from (2.12) and (2.13) that

∥un∥2−∫Bun2∗dx−μ∫Bf(x,un)undx+|DJ|(un)∥un∥=0. (2.14)

Since f(x, u_n)u_n≤1 for all n, by the dominated convergence theorem, one gets

∫Bf(x,un)undx→∫Bf(x,u)udx.

Set w_n=u_n−u and lim_{n → ∞}∣w_n∣=l > 0, by Brézis-Lieb’s lemma, Lemma 2.1 and (2.14), we have

∥wn∥2+∥u∥2−∫Bwn2∗dx−∫Bu2∗dx−μ∫Bf(x,u)udx=o(1). (2.15)

On the other hand, for φ ∈ P, taking v=u_n+φ in (2.11) and letting n → ∞, by Fatou’s lemma, we obtain

μ∫Bf(x,u)φdx≤∫B∇u∇φdx−∫Bu2∗−1φdx, for φ∈P. (2.16)

Denote unT=min{un,T} , T > 0. Taking v=un−unT∈P in (2.11), one has

−μ∫Bf(x,un)unTdx≤−∫B∇un∇unTdx+∫Bun2∗−1unTdx+|DJ|(un)∥unT∥.

Taking the limit n → ∞ first, then T → ∞, one gets

−μ∫Bf(x,u)udx≤−∫B|∇u|2dx+∫Bu2∗dx. (2.17)

It follows from (2.16) and (2.17) that

μ∫Bf(x,u)(φ−u)dx≤∫B∇u∇(φ−u)dx−∫Bu2∗−1(φ−u)dx, for φ∈P.

Therefore, similar to the proof of (i) in Lemma 2.3, for all φ∈H01(B) , we have

∫B∇u∇φdx−∫Bu2∗−1φdx−μ∫Bf(x,u)φdx=0.

In particular, one has

∥u∥2−∫Bu2∗dx−μ∫Bf(x,u)udx=0. (2.18)

It follows from (2.15) and (2.18) that

∥wn∥2−∫Bwn2∗dx=o(1). (2.19)

Consequently, it follows from (2.10) that l2≥SN2 . Then, by (2.2), (2.3), (2.18) and the Young inequality, one has

J(u)≥1N∥u∥2−μ∫BF(x,u+)dx≥1N∥u∥2−Cμ≥−Θμ,

where Θ=Θ(N,S,|B|,∫B1ϕ1dx) . Besides, by Lemma 2.1, (2.19) and the condition c<1NSN2−Θμ , one has

J(u)=J(un)−12∥wn∥2+12∗∫Bwn2∗dx+o(1)≤c−1Nl2<1NSN2−Θμ−1NSN2=−Θμ,

which contradicts the above inequality. Therefore, l=0, which implies that u_n → u in H01(B) . This completes the proof of Lemma 2.4. □

Now, we show that the functional J satisfies the Mountain-pass lemma.

Lemma 2.5

There exist constants r, ρ, μ₀ > 0 such that the functional J satisfies the following conclusions:

(i) J|u∈Sρ≥r>0 and infu∈BρJ(u)<0 for μ ∈ (0, μ₀).

(ii) There exists e∈H01(B) such that ∣e∣>ρ and J(e) < 0.

Proof

(i) By (2.2) and (2.3), one has

J(u)≥12∥u∥2−S−2∗22∗∥u∥2∗−Cμ.

We see that there exist constants ρ, r, Λ₀ > 0, such that J(u)|Sρ≥r for every μ ∈ (0, μ₀). Moreover, for u∈H01(B)∖{0} , it holds

limt→0+J(tu)t=−μlimt→0+∫B∫0tu+f(x,s)dstdx=−μ∫Bf(x,0)u+dx=−μ∫Bu+wμdx<0.

So we obatin J (tu)<0 for all u⁺≠0 and t small enough. Therefore, for ∣u∣ small enough, one has

d≜infu∈BρJ(u)<0. (2.20)

(ii) For every u∈H01(B),u+≠0 , we have J (tu) → −∞ as t → +∞. Thus, there is e∈H01(B) such that ∣e∣>ρ and J(e) < 0. The proof is completed. □

According to Lemma 2.1, similar to [16], we can easy obtain that d is attained at some u^⋆ ∈ B_ρ. According to Lemma 2.3 and Lemma 2.5 (i), we obtain the following result.

Theorem 2.6

For 0<μ<μ₀, problem (1.1) has a positive solution u^⋆ with J(u^⋆)=d < 0.

Denote

Uε(x)=[N(N−2)]N−24εN−22(ε2+|x|2)N−22, x∈RN, ε>0.

Then, the infimum (2.10) is attained at U(x). Choosing η∈C0∞(Bδ(x0),[0,1]) such that η(x)=1 near x=x₀, where B_δ(x₀)⊂ B. Let φ_ε=U_εη, we have the following conclusion.

Lemma 2.7

There holds supt≥0J(u∗+tφε)≤1NSN2−Θμ for sufficient small μ>0.

Proof

From [17], we have

∫B|∇φε|2dx=SN2+O(εN−2),∫Bφε2⋆dx=SN2+O(εN),∫Bφε2⋆−1dx=cεN−22, (2.21)

where c > 0 is a constant. Since J(u^⋆+tφ_ε) → −∞ as t → ∞, J(u^⋆)<0, according to Lemma 2.5 (i), we can assume that there exist t₁, t₂ > 0 such that sup_t≥0J(u^⋆+tφ_ε)=sup_{t∈[t₁, t₂]}J(u^⋆+tφ_ε).

Since u^⋆ is a positive solution of problem (1.1), we have

{ J(u⋆)<0,∫B∇u⋆∇φεdx=∫B(u⋆)2⋆−1φεdx+μ∫Bφεu⋆dx,φeu⋆∈L1(B), that is ∫Bφεu⋆dx≤C for some C>0. (2.22)

Moreover, according to u^⋆^>w_μ, we deduce that

F(x,u⋆)=∫0wμ1wμdt+∫wμu⋆1tdt=1+lnu⋆−lnwμ.

Consequently, one has

∫B[F(x,u⋆+tφε)-F(x,u⋆)]dx=∫B[1+ln(u⋆+tφε)-lnwμ]dx-∫B(1+lnu⋆-lnwμ)dx=∫Bln1+tφεu⋆dx≥0 (2.23)

for all t≥0. Therefore, from (2.22), (2.23) and the following inequality

(a+b)2⋆≥a2⋆+2⋆a2⋆−1b+2⋆ab2⋆−1+b2⋆, for a,b≥0,

we have

J(u⋆+tφε)=12‖u⋆+tφε‖2-12⋆∫B(u⋆+tφε)2⋆dx-μ∫BF(x,u⋆+tφε)dx=12‖u⋆‖2+t22‖φε‖2+t∫B∇u⋆∇φεdx-12⋆∫B(u⋆+tφε)2⋆dx-μ∫BF(x,u⋆+tφε)dx=J(u⋆)+t22‖φε‖2-12⋆∫B[(u⋆+tφε)2⋆-(u⋆)2⋆-2⋆(u⋆)2⋆-1tφε]dx-μ∫B[F(x,u⋆+tφε)-F(x,u⋆)]dx+tμ∫Bφεu⋆dx≤t22‖φε‖2-t2⋆2⋆∫Bφε2⋆dx-t2⋆-1∫Bu⋆φε2⋆-1dx+tμ∫Bφεu⋆dx. (2.24)

For t=0, let

g(t)=t22‖φε‖2−t2⋆2⋆∫Bφε2⋆dx.

Then, we have g′(t)=t[ ‖φε‖2−t2⋆−2∫Bφε2⋆dx ] . Let g′(t)=0, one has

tmax=(‖φε‖2∫Bφε2⋆dx)12⋆−2

such that g′(t) > 0 for 0 < t<t_max and g′(t) < 0 for t > t_max. Moreover, g(t_max)=max_t≥0g(t). By a standard regularity argument, one has u^⋆ ∈ C¹(B, ℝ⁺) and there exists a positive constant C (independent of x) such that u^⋆<C. Consequently, it follows from (2.21)-(2.24) that

supt∈[t1,t2]J(u⋆+tφε)≤supt≥0(t22‖φε‖2−t2⋆2⋆∫Bφε2⋆dx)−Ct12⋆−1∫Bφε2⋆−1dx+t2μ∫Bφεu⋆dx≤g(tmax)+C1μ−C2εN−22=1NSN2+C3εN−2+C1μ−C2εN−22.

Let ε=μ1N−2 , μ1=min{1,(C2C3+C1+Θ)2} , we have

C3εN−2+C1μ−C2εN−22=(C3+C1)μ−C2μ12=μ(C3+C1−C2μ−12)≤−Θμ

for all μ ∈ (0, μ₁). This leads us to the proof of Lemma 2.7. □

Theorem 2.8

There exists µ_⋆>0 such that for 0<μ<µ_⋆, problem (1.1) has at least two positive solutions.

Proof

Let 0<μ<µ_⋆=min{μ₀, μ₁}. By Lemma 2.4 and Lemma 2.7, J satisfies the Palais-Smale condition at the level c. By Lemma 2.5, there exists a Palais-Smale sequence {u_n} such that ∣DJ∣(u_n) → 0, J(u_n) → c as n → ∞. Up to a subsequence, u_n → v^⋆ in H01(B) , and J(v^⋆)=lim_{n → ∞}J(u_n)=c > 0, ∣DJ∣(u_n) → 0. Applying the Mountain pass lemma [14] and Lemma 2.3, v^⋆ is a positive solution of problem (1.1). Combining with Theorem 2.6, the proof of Theorem 2.8 is completed. □

2.2 Radially Symmetric and Monotone Decreasing Solution

In this part, we shall prove that every positive solution u of problem (1.1) is radially symmetric and monotone decreasing about the origin. Let u=μ12v , then we deduce that v satisfies the following equation

{ Δv+μ2⋆−22v2⋆−1+1v=0, in B, v=0, on ∂B. (2.25)

Therefore, it is sufficient to prove that v is radially symmetric and monotone decreasing about the origin. First, we introduce some notations.

Choose any direction to be the x₁ direction. Let

Tλ={x∈ℝn|x1=λ, for some λ∈ℝ}

be the moving plane, and the set

Σλ={x∈B1(0)|x1<λ}

be the region to the left of the plane. We use the standard notation

xλ=(2λ−x1,x2,…,xn)

for the reflection of x about the plane T_λ.

We denote v(x^λ)=v_λ(x), then compare the values of v(x) and v_λ(x), let

wλ(x)=vλ(x)−v(x).

Otherwise, after a direct calculation, we derive that w_λ(x^λ)=−w_λ(x), hence it is said to be anti-symmetric.

We carry out the proof in two steps. To begin with, we show that for λ sufficiently close to −1, we have

wλ(x)≥0,x∈Σλ. (2.26)

This provides a starting point to move the plane T_λ.

Next, we move the plane T_λ along the x₁ direction to the right as long as inequality (2.26) holds. The plane T_λ will eventually stop at some limiting position λ₀, where

λ0=sup{λ|wρ(x)≥0,ρ≤λ},

then we are able to claim that

wλ0(x)≡0,x∈Σλ0.

The symmetry and monotonicity of solution v about T_λ follow naturally from the proof. Also, because of the arbitrariness of the x₁ direction, we conclude that v must be radially symmetric and monotone decreasing about the origin.

Step 1. We show that for λ sufficiently close to −1, we have

wλ(x)≥0,x∈Σλ. (2.27)

If not, we denote there exists some point x₀ ∈ Σ_λ, such that

wλ(x0)=minΣλwλ(x)<0.

By problem (2.25) and the Mean Value Theorem, we can easy derive that

− Δ w λ x 0 = − Δ v λ x 0 − − Δ v x 0 = 1 v λ x 0 + μ 2 ∗ − 2 2 v λ 2 ∗ − 1 x 0 − 1 v x 0 + μ 2 ∗ − 2 2 v 2 ∗ − 1 x 0 = v λ − 1 x 0 − v − 1 x 0 + μ 2 ∗ − 2 2 v λ 2 ∗ − 1 x 0 − v 2 ∗ − 1 x 0 = − ξ λ − 2 x 0 w λ x 0 + 2 ⋆ − 1 μ 2 ∗ − 2 2 η λ 2 ∗ − 2 x 0 w λ x 0 ≥ − v − 2 x 0 + 2 ∗ − 1 μ 2 ∗ − 2 2 v 2 ∗ − 2 x 0 w λ x 0 , (2.28)

where ξ_λ(x₀), η_λ(x₀) are in between v_λ(x₀) and v(x₀), and the last inequality is due to the negative point x₀ of w_λ(x). That is

vλ(x0)<v(x0),

we have

0<vλ(x0)≤ξλ(x0),ηλ(x0)≤v(x0).

Then we obtain that

−Δwλ(x0)+c(x0)wλ(x0)≥0, (2.29)

where

c(x0)=v−2(x0)−(2∗−1)μ2⋆−22v2∗−2(x0)=1−(2∗−1)μ2⋆−22v2∗(x0)v2(x0).

Since v ∈ C^∞(B) ∩ L^∞(B) (see [4]), v(x₀) is bounded. We can see that c(x₀)>0 provided μ enough small.

On the other hand, at the minmum point, we have that

−Δwλ(x0)+c(x0)wλ(x0)<0,

which contradicts with (2.29). Therefore (2.27) hods. This completes the preparation for the moving of planes.

Step 2. Inequality (2.27) provides a starting point, from which we move the plane T_λ toward the right as long as (2.27) holds to its limiting position to show that v is monotone decreasing about the origin. More precisely, we define

λ0=sup{λ|wρ(x)≥0,ρ≤λ}.

We will show that

λ0=0, and wλ0(x)≡0. (2.30)

If not, then λ₀ < 0, we show that the plane can be moved further right to cause a contradiction with the definition of λ₀. More precisely, there exists a small ε>0 such that for all λ ∈ (λ₀, λ₀+ε) such that

wλ(x)≥0,x∈Σλ, (2.31)

which contradicts the definition of λ₀. Hence, (2.30) must be true.

Now we need to prove (2.31) is true. First, by the definition of λ₀, we have wλ0(x)≥0 in Σλ0 . Moreover, by the strong maximum principle, we obtain

wλ0(x)>0,x∈Σλ0.

We claim that there exists a constant C₀ such that

wλ0(x)≥C0>0,x∈Σλ0−δ.

Due to the continuity of wλ0(x) with respect to λ, for sufficiently small ϵ and any λ ∈ (λ₀, λ₀+ε), one obtains

wλ(x)≥0,x∈Σλ0−δ. (2.32)

Denote Ω_λ=Σ_λ\Σ_λ₀−δ this is a narrow region, we need to prove that w_λ satisfies the conditions of narrow region principle. From the (2.28), we know that, for any x ∈ Ω_λ,

− Δ w λ ( x ) = − Δ v λ ( x ) − ( − Δ v ( x ) ) = 1 v λ ( x ) + μ 2 ∗ − 2 2 v λ 2 ∗ − 1 ( x ) − 1 v ( x ) + μ 2 ∗ − 2 2 v 2 ∗ − 1 ( x ) = v λ − 1 ( x ) − v − 1 ( x ) + μ 2 ∗ − 2 2 v λ 2 ∗ − 1 ( x ) − v 2 ∗ − 1 ( x ) = − ξ λ − 2 ( x ) w λ ( x ) + 2 ⋆ − 1 μ 2 ∗ − 2 2 η λ 2 ∗ − 2 ( x ) w λ ( x ) = − ξ λ 2 ( x ) + 2 ⋆ − 1 μ 2 ∗ − 2 2 η λ 2 ∗ − 2 ( x ) w λ ( x ) ,

where ξ_λ(x), η_λ(x) are in between v_λ(x) and v(x). Consequently, one has

−Δwλ(x)+c(x)wλ(x)≥0,x∈Ωλ,

where c(x)=ξλ−2(x)−(2∗−1)μ2⋆−22ηλ2∗−2(x) . Noting that v ∈ C^∞(B) ∩ L^∞(B), we have c(x) is bounded in Ω_λ. So w_λ satisfies the narrow region principle

{ −Δwλ(x)+c(x)wλ(x)≥0,x∈Ωλ,wλ(x)≥0,x∈∂Ωλ, (2.33)

then we derive that

wλ(x)≥0,x∈Ωλ.

Combining with (2.32), we have

wλ(x)≥0,x∈Σλ.

This (2.31) is true. Therefore, we must have

λ0=0.

Further more, we have

w0(x)≥0,x∈Σ0.

If we move the plane from λ is sufficiently close to 1, then we move the plane T_λ along the x₁ direction to the left. By a similar argument, we can derive

w0(x)≤0,x∈Σ0,

then we have

w0(x)≡0,x∈Σ0.

We drive that v(x) is symmetric about the plane T₀. Moreover, the arbitrariness of the x₁ direction leads to the radial symmetry of v(x) about the origin, so does u. The monotonicity comes directly from the argument. This completes the proof.

Acknowledgements

The authors express their gratitude to the reviewer for careful reading and helpful suggestions which led to an improvement of the original manuscript.

Supported by Science and Technology Foundation of Guizhou Province(KJ[2019]1163); Fundamental Research Funds of China West Normal University(18B015); Innovative Research Team of China West Normal University(CXTD2018-8).

References

[1] A.C. Lazer and P.J. Mckenna, On a singular nonlinear elliptic boundary-value problem, Proc. Amer. Math. Soci. 111 (1991), no. 3, 721–730.10.1090/S0002-9939-1991-1037213-9Search in Google Scholar

[2] M. Ghergu and V. Rǎulescu, Multi-parameter bifurcation and asymptotics for the singular Lane-Emden-Fowler equation with a convection term, Proc. Roy. Soc. Edinburgh Sect. A 135 (2005), no. 1, 61–83.10.1017/S0308210500003760Search in Google Scholar

[3] J. Giacomoni and K. Saoudi, Multiplicity of positive solutions for a singular and critical problem, Nonlinear Anal. 71 (2009), no. 9, 4060–4077.10.1016/j.na.2009.02.087Search in Google Scholar

[4] N. Hirano, C. Saccon and N. Shioji, Existence of multiple positive solutions for singular elliptic problems with concave and convex nonlinearities, Adv. Differential Equations 9 (2004), no. 1-2, 197–220.Search in Google Scholar

[5] N. Hirano, C. Saccon and N. Shioji, Brezis-Nirenberg type theorems and multiplicity of positive solutions for a singular elliptic problem, J. Differential Equations 245 (2008), no. 8, 1997–2037.10.1016/j.jde.2008.06.020Search in Google Scholar

[6] J.F. Liao, J. Liu and P. Zhang, C.L. Tang, Existence of two positive solutions for a class of semilinear elliptic equations with singularity and critical exponents, Ann. Polon. Math. 116 (2016), no. 3, 273–292.10.4064/ap3606-10-2015Search in Google Scholar

[7] Y.J. Sun and S.P. Wu, An exact estimate result for a class of singular equations with critical exponents, J. Funct. Anal. 260 (2011), no. 5, 1257–1284.10.1016/j.jfa.2010.11.018Search in Google Scholar

[8] Y.J. Sun, S.P. Wu and Y.M. Long, Combined effects of singular and superlinear nonlinearities in some singular boundary value problems, J. Differential Equations 176 (2001), no. 2, 511–531.10.1006/jdeq.2000.3973Search in Google Scholar

[9] X. Wang, X.Q. Qin and G. Hu, Existence of weak positive solution for a singular elliptic problem with supercritical nonlinearity, Anal. Math. Phys. 8 (2018), no. 1, 43–55.10.1007/s13324-016-0162-4Search in Google Scholar

[10] X. Wang, L. Zhao and P.H. Zhao, Combined effects of singular and critical nonlinearities in elliptic problems, Nonlinear Anal. 87 (2013), 1–10.10.1016/j.na.2013.03.019Search in Google Scholar

[11] H.T. Yang, Multiplicity and asymptotic behavior of positive solutions for a singular semilinear elliptic problem, J. Differential Equations 189 (2003), no. 2, 487–512.10.1016/S0022-0396(02)00098-0Search in Google Scholar

[12] Y.J. Sun, Compatibility phenomena in singular problems, Proc. Roy. Soc. Edinburgh Sect. A 143 (2013), no. 6, 1321–1330.10.1017/S030821051100117XSearch in Google Scholar

[13] Y.J. Sun and D.Z. Zhang, The role of the power 3 for elliptic equations with negative exponents, Calc. Var. Partial Differential Equations 49 (2014), no. 3-4, 909–922.10.1007/s00526-013-0604-xSearch in Google Scholar

[14] A. Canino and M. Degiovanni, Nonsmooth critical point theory and quasilinear elliptic equations, in: Topological Methods in Differential Equations and Inclsions (Montré, 1994), in: NATO ASI series, C, Vol. 472, Kluwer, Dordrecht, 1995, 1–50.10.1007/978-94-011-0339-8_1Search in Google Scholar

[15] X.Q. Liu, Y.X. Guo and J.Q. Liu, Solutions for singular p-Laplace equation in ℝ^N, Jrl Syst. Sci. Complexity 22 (2009), no. 4, 597–613.10.1007/s11424-009-9190-6Search in Google Scholar

[16] C.Y. Lei, J.F. Liao, C.L. Tang, Multiple positive solutions for Kirchhoff type of problems with singularity and critical exponents, J. Math. Anal. Appl. 421 (2015), no. 1, 521–538.10.1016/j.jmaa.2014.07.031Search in Google Scholar

17 H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent, Comm. Pure. Appl. Math. 36 (1983) 437–477.10.1002/cpa.3160360405Search in Google Scholar

Received: 2020-08-11

Accepted: 2021-02-01

Published Online: 2021-04-14

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

Positive radial symmetric solutions for a class of elliptic problems with critical exponent and -1 growth

Abstract

1 Introduction and main result

Theorem 1.1

Remark 1

2 Proof of Theorem 1.1

2.1 Existence of Two Positive Solutions

Lemma 2.1

Proof

Lemma 2.2

Proof

Lemma 2.3

Proof

Lemma 2.4

Proof

Lemma 2.5

Proof

Theorem 2.6

Lemma 2.7

Proof

Theorem 2.8

Proof

2.2 Radially Symmetric and Monotone Decreasing Solution

Acknowledgements

References

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