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Advances in Nonlinear Analysis

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A class of semipositone p-Laplacian problems with a critical growth reaction term

Kanishka Perera
  • Corresponding author
  • Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA
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/ Ratnasingham Shivaji / Inbo Sim
Published Online: 2019-06-16 | DOI: https://doi.org/10.1515/anona-2020-0012

Abstract

We prove the existence of ground state positive solutions for a class of semipositone p-Laplacian problems with a critical growth reaction term. The proofs are established by obtaining crucial uniform C1,α a priori estimates and by concentration compactness arguments. Our results are new even in the semilinear case p = 2.

Keywords: critical semipositone p-Laplacian problems; ground state positive solutions; concentration compactness; uniform C1,α a priori estimates

MSC 2010: Primary 35B33; Secondary 35J92; 35B09; 35B45

1 Introduction

Consider the p-superlinear semipositone p-Laplacian problem

Δpu=uq1μin Ωu>0in Ωu=0on Ω,(1.1)

where Ω is a smooth bounded domain in ℝN, 1 < p < N, p < qp, μ > 0 is a parameter, and p = Np/(Np) is the critical Sobolev exponent. The scaling uμ1/(q–1) u transforms the first equation in (1.1) into

Δpu=μ(qp)/(q1)uq11,

so in the subcritical case q < p, it follows from the results in Castro et al. [1] and Chhetri et al. [2] that this problem has a weak positive solution for sufficiently small μ > 0 when p > 1 (see also Unsurangie [3], Allegretto et al. [4], Ambrosetti et al. [5], and Caldwell et al. [6] for the case when p = 2). On the other hand, in the critical case q = p, it follows from a standard argument involving the Pohozaev identity for the p-Laplacian (see Guedda and Véron [7, Theorem 1.1]) that problem (1.1) has no solution for any μ > 0 when Ω is star-shaped. The purpose of the present paper is to show that this situation can be reversed by the addition of lower-order terms, as was observed in the positone case by Brézis and Nirenberg in the celebrated paper [8]. However, this extension to the semipositone case is not straightforward as u = 0 is no longer a subsolution, making it much harder to find a positive solution as was pointed out in Lions [9]. The positive solutions that we obtain here are ground states, i.e., they minimize the energy among all positive solutions.

We study the Brézis-Nirenberg type critical semipositone p-Laplacian problem

Δpu=λup1+up1μin Ωu>0in Ωu=0on Ω,(1.2)

where λ, μ > 0 are parameters. Let W01,p(Ω) be the usual Sobolev space with the norm given by

up=Ω|u|pdx.

For a given λ > 0, the energy of a weak solution uW01,p(Ω) of problem (1.2) is given by

Iμ(u)=Ω(|u|ppλuppupp+μu)dx,

and clearly all weak solutions lie on the set

Nμ=uW01,p(Ω):u>0 in Ω and Ω|u|pdx=Ωλup+upμudx.

We will refer to a weak solution that minimizes Iμ on 𝓝μ as a ground state. Let

λ1=infuW01,p(Ω)0Ω|u|pdxΩ|u|pdx(1.3)

be the first Dirichlet eigenvalue of the p-Laplacian, which is positive. We will prove the following existence theorem.

Theorem 1.1

If Np2 and λ ∈ (0, λ1), then there exists μ > 0 such that for all μ ∈ (0, μ), problem (1.2) has a ground state solution uμC1,α(Ω) for some α ∈ (0, 1).

The scaling uμ–1/(pp) u transforms the first equation in the critical semipositone p-Laplacian problem

Δpu=λup1+μup11in Ωu>0in Ωu=0on Ω(1.4)

into

Δpu=λup1+up1μ(p1)/(pp),

so as an immediate corollary we have the following existence theorem for problem (1.4).

Theorem 1.2

If Np2 and λ ∈ (0, λ1), then there exists μ > 0 such that for all μ ∈ (0, μ), problem (1.4) has a ground state solution uμC1,α(Ω) for some α ∈ (0, 1).

We would like to emphasize that Theorems 1.1 and 1.2 are new even in the semilinear case p = 2.

The outline of the proof of Theorem 1.1 is as follows. We consider the modified problem

Δpu=λu+p1+u+p1μf(u)in Ωu=0on Ω,(1.5)

where u+(x) = max {u(x), 0} and

f(t)=1,t01|t|p1,1<t<00,t1.

Weak solutions of this problem coincide with critical points of the C1-functional

Iμ(u)=Ω(|u|ppλu+ppu+pp)dx+μ[u0udx+1<u<0u|u|p1updx11pu1],uW01,p(Ω),

where |⋅| denotes the Lebesgue measure in ℝN. Recall that Iμ satisfies the Palais-Smale compactness condition at the level c ∈ ℝ, or the (PS)c condition for short, if every sequence (uj) ⊂ W01,p(Ω) such that Iμ(uj) → c and Iμ(uj) → 0, called a (PS)c sequence for Iμ, has a convergent subsequence. As we will see in Lemma 2.1 in the next section, it follows from concentration compactness arguments that Iμ satisfies the (PS)c condition for all

c<1NSN/p11pμΩ,

where S is the best Sobolev constant (see (2.1)). First we will construct a mountain pass level below this threshold for compactness for all sufficiently small μ > 0. This part of the proof is more or less standard. The novelty of the paper lies in the fact that the solution uμ of the modified problem (1.5) thus obtained is positive, and hence also a solution of our original problem (1.2), if μ is further restricted. Note that this does not follow from the strong maximum principle as usual since –μ f(0) < 0. This is precisely the main difficulty in finding positive solutions of semipositone problems (see Lions [9]). We will prove that for every sequence μj → 0, a subsequence of uμj is positive in Ω. The idea is to show that a subsequence of uμj converges in C01(Ω) to a solution of the limit problem

Δpu=λup1+up1in Ωu>0in Ωu=0on Ω.

This requires a uniform C1,α(Ω) estimate of uμj for some α ∈ (0, 1). We will obtain such an estimate by showing that uμj is uniformly bounded in W01,p(Ω) and uniformly equi-integrable in Lp(Ω), and applying a result of de Figueiredo et al. [10]. The proof of uniform equi-integrability in Lp(Ω) involves a second (nonstandard) application of the concentration compactness principle. Finally, we use the mountain pass characterization of our solution to show that it is indeed a ground state.

Remark 1.3

Establishing the existence of solutions to the critical semipositone problem

Δpu=μup1+up11inΩu>0inΩu=0onΩ

for small μ remains open.

2 Preliminaries

Let

S=infuW01,p(Ω)0Ω|u|pdxΩ|u|pdxp/p(2.1)

be the best constant in the Sobolev inequality, which is independent of Ω. The proof of Theorem 1.1 will make use of the following compactness result.

Lemma 2.1

For any fixed λ, μ > 0, Iμ satisfies the (PS)c condition for all

c<1NSN/p11pμΩ.(2.2)

Proof

Let (uj) be a (PS)c sequence. First we show that (uj) is bounded. We have

Iμ(uj)=Ω(|uj|ppλuj+ppuj+pp)dx+μ[uj0ujdx+1<uj<0uj|uj|p1ujpdx11puj1]=c+o(1)(2.3)

and

Iμ(uj)v=Ω|uj|p2ujvλuj+p1vuj+p1vdx+μ[uj0vdx+1<uj<01|uj|p1vdx]=o(1)vvW01,p(Ω).(2.4)

Taking v = uj in (2.4), dividing by p, and subtracting from (2.3) gives

1NΩuj+pdxc+11pμΩ+o(1)uj+1,(2.5)

and it follows from this, (2.3), and the Hölder inequality that (uj) is bounded in W01,p(Ω).

Since (uj) is bounded, so is (uj+), a renamed subsequence of which then converges to some v ≥ 0 weakly in W01,p(Ω), strongly in Lq(Ω) for all q ∈ [1, p) and a.e. in Ω, and

|uj+|pdxwκ,uj+pdxwν(2.6)

in the sense of measures, where κ and ν are bounded nonnegative measures on Ω (see, e.g., Folland [11]). By the concentration compactness principle of Lions [12, 13], then there exist an at most countable index set I and points xiΩ, iI such that

κ|v|pdx+iIκiδxi,ν=vpdx+iIνiδxi,(2.7)

where κi, νi > 0 and νip/pκi/S. We claim that I = ∅. Suppose by contradiction that there exists iI. Let φ : ℝN → [0, 1] be a smooth function such that φ(x) = 1 for |x| ≤ 1 and φ(x) = 0 for |x| ≥ 2. Then set

φi,ρ(x)=φxxiρ,xRN

for iI and ρ > 0, and note that φi,ρ : ℝN → [0, 1] is a smooth function such that φi,ρ(x) = 1 for |xxi| ≤ ρ and φi,ρ(x) = 0 for |xxi| ≥ 2 ρ. The sequence (φi,ρ uj+) is bounded in W01,p(Ω) and hence taking v = φi,ρ uj+ in (2.4) gives

Ω(φi,ρ|uj+|p+uj+|uj+|p2uj+φi,ρλφi,ρuj+pφi,ρuj+p+μφi,ρuj+)dx=o(1).(2.8)

By (2.6),

Ωφi,ρ|uj+|pdxΩφi,ρdκ,Ωφi,ρuj+pdxΩφi,ρdν.

Denoting by C a generic positive constant independent of j and ρ,

Ω(uj+|uj+|p2uj+φi,ρλφi,ρuj+p+μφi,ρuj+)dxC1ρ+μIj1/p+Ij,

where

Ij:=ΩB2ρ(xi)uj+pdxΩB2ρ(xi)vpdxCρpΩB2ρ(xi)vpdxp/p.

So passing to the limit in (2.8) gives

Ωφi,ρdκΩφi,ρdνC(1+μρ)ΩB2ρ(xi)vpdx1/p+ΩB2ρ(xi)vpdx.

Letting ρ ↘ 0 and using (2.7) now gives κiνi, which together with νi > 0 and νip/pκi/S then gives νiSN/p. On the other hand, passing to the limit in (2.5) and using (2.6) and (2.7) gives

νiNc+11pμΩ<SN/p

by (2.2), a contradiction. Hence I = ∅ and

Ωuj+pdxΩvpdx.(2.9)

Passing to a further subsequence, uj converges to some u weakly in W01,p(Ω), strongly in Lq(Ω) for all q ∈ [1, p), and a.e. in Ω. Since

|uj+p1(uju)|uj+p+uj+p1|u|21puj+p+1p|u|p

by Young’s inequality,

Ωuj+p1(uju)dx0

by (2.9) and the dominated convergence theorem. Then taking v = uju in (2.4) gives

Ω|uj|p2uj(uju)dx0,

so uju in W01,p(Ω) for a renamed subsequence (see, e.g., Perera et al. [14, Proposition 1.3]).□

The infimum in (2.1) is attained by the family of functions

uε(x)=CN,pε(Np)/p2(ε+|x|p/(p1))(Np)/p,ε>0

when Ω = ℝN, where the constant CN,p > 0 is chosen so that

RN|uε|pdx=RNuεpdx=SN/p.

Without loss of generality, we may assume that 0 ∈ Ω. Let r > 0 be so small that B2r(0) ⊂ Ω, take a function ψC0(B2r(0), [0, 1]) such that ψ = 1 on Br(0), and set

u~ε(x)=ψ(x)uε(x),vε(x)=u~ε(x)Ωu~εpdx1/p,

so that Ωvεpdx=1. Then we have the well-known estimates

Ω|vε|pdxS+Cε(Np)/p,(2.10)

Ωvεpdx1Cεp1,N>p21Cεp1|logε|,N=p2,(2.11)

where C = C(N, p) > 0 is a constant (see, e.g., Drábek and Huang [15]).

3 Proof of Theorem 1.1

First we show that Iμ has a uniformly positive mountain pass level below the threshold for compactness given in Lemma 2.1 for all sufficiently small μ > 0. Let vε be as in the last section.

Lemma 3.1

There exist μ0, ρ, c0 > 0, R > ρ, and β < 1N SN/p such that the following hold for all μ ∈ (0, μ0):

  1. u∥ = ρIμ(u) ≥ c0,

  2. Iμ(tvε) ≤ 0 for all tR and ε ∈ (0, 1],

  3. denoting by Γ = {yC([0, 1], W01,p(Ω)) : y(0) = 0, y(1) = Rvε} the class of paths joining the origin to Rvε,

    c0cμ:=infyΓmaxuy([0,1])Iμ(u)β11pμΩ(3.1)

    for all sufficiently small ε > 0,

  4. Iμ has a critical point uμ at the level cμ.

Proof

By (1.3) and (2.1),

Iμ(u)1p1λλ1upSp/ppup11pμΩ,

and (i) follows from this for sufficiently small ρ, c0, μ > 0 since λ < λ1.

Since vε ≥ 0,

Iμ(tvε)=tppΩ(|vε|pλvεp)dxtpp+μtΩvεdx

for t ≥ 0. By the Hölder’s and Young’s inequalities,

μtΩvεdxμtΩ11/pΩvεpdx1/pCλμp/(p1)+λtp2pΩvεpdx,

where

Cλ=11p2λ1/(p1)Ω,

so

Iμ(tvε)tppΩ|vε|pλ2vεpdxtpp+Cλμp/(p1).(3.2)

Then by (2.10) and for ε, μ ∈ (0, 1],

Iμ(tvε)(S+C)tpptpp+Cλ,

from which (ii) follows for sufficiently large R > ρ.

The first inequality in (3.1) is immediate from (i) since R > ρ. Maximizing the right-hand side of (3.2) over t ≥ 0 gives

cμ1NΩ|vε|pλ2vεpdxN/p+Cλμp/(p1),

and (2.10) and (2.11) imply that the integral on the right-hand side is strictly less than S for all sufficiently small ε > 0 since Np2 and λ > 0, so the second inequality in (3.1) holds for sufficiently small μ > 0.

Finally, (iv) follows from (i)–(iii), Lemma 2.1, and the mountain pass lemma (see Ambrosetti and Rabinowitz [16]).□

Next we show that uμ is uniformly bounded in W01,p(Ω) and uniformly equi-integrable in Lp(Ω), and hence also uniformly bounded in C1,α(Ω) for some α ∈ (0, 1) by de Figueiredo et al. [10, Proposition 3.7], for all sufficiently small μ ∈ (0, μ0).

Lemma 3.2

There exists μ ∈ (0, μ0] such that the following hold for all μ ∈ (0, μ):

  1. uμ is uniformly bounded in W01,p(Ω),

  2. E|uμ|pdx0asE0, uniformly in μ,

  3. uμ is uniformly bounded in C1,α(Ω) for some α ∈ (0, 1).

Proof

We have

Iμ(uμ)=Ω(|uμ|ppλuμ+ppuμ+pp)dx+μ[uμ0uμdx+1<uμ<0uμ|uμ|p1uμpdx11puμ1]=cμ(3.3)

and

Iμ(uμ)v=Ω|uμ|p2uμvλuμ+p1vuμ+p1vdx+μ[uμ0vdx+1<uμ<01|uμ|p1vdx]=0vW01,p(Ω).(3.4)

Taking v = uμ in (3.4), dividing by p, and subtracting from (3.3) gives

1NΩuμ+pdxcμ+11pμΩβ(3.5)

by (3.1), and (i) follows from this, (3.4) with v = uμ, and the Hölder inequality.

If (ii) does not hold, then there exist sequences μj → 0 and (Ej) with |Ej| → 0 such that

lim_Ej|uμj|pdx>0.(3.6)

Since (uμj) is bounded by (i), so is (uμj+), a renamed subsequence of which then converges to some v ≥ 0 weakly in W01,p(Ω), strongly in Lq(Ω) for all q ∈ [1, p) and a.e. in Ω, and

|uμj+|pdxwκ,uμj+pdxwν(3.7)

in the sense of measures, where κ and ν are bounded nonnegative measures on Ω. By Lions [12, 13], then there exist an at most countable index set I and points xiΩ, iI such that

κ|v|pdx+iIκiδxi,ν=vpdx+iIνiδxi,(3.8)

where κi, νi > 0 and νip/pκi/S. Suppose I is nonempty, say, iI. An argument similar to that in the proof of Lemma 2.1 shows that κiνi, so νiSN/p. On the other hand, passing to the limit in (3.5) with μ = μj and using (3.7) and (3.8) gives νi < SN/p, a contradiction. Hence I = ∅ and

Ωuμj+pdxΩvpdx.

As in the proof of Lemma 2.1, a further subsequence of (uμj) then converges to some u in W01,p(Ω), and hence also in Lp(Ω), and a.e. in Ω. Then

Ej|uμj|pdxΩ|uμj|p|u|pdx+Ej|u|pdx0,

contradicting (3.6).

Finally, (iii) follows from (i), (ii), and de Figueiredo et al. [10, Proposition 3.7].□

We are now ready to prove Theorem 1.1.

Proof of Theorem 1.1

We claim that uμ is positive in Ω, and hence a weak solution of problem (1.2), for all sufficiently small μ ∈ (0, μ). It suffices to show that for every sequence μj → 0, a subsequence of uμj is positive in Ω. By Lemma 3.2 (iii), a renamed subsequence of uμj converges to some u in C01(Ω). We have

Iμj(uμj)=Ω(|uμj|ppλuμj+ppuμj+pp)dx+μj[uμj0uμjdx+1<uμj<0uμj|uμj|p1uμjpdx11puμj1]=cμjc0

by (3.1) and

Iμj(uμj)v=Ω|uμj|p2uμjvλuμj+p1vuμj+p1vdx+μj[uμj0vdx+1<uμj<01|uμj|p1vdx]=0vW01,p(Ω),

and passing to the limits gives

Ω(|u|ppλu+ppu+pp)dxc0

and

Ω|u|p2uvλu+p1vu+p1vdx=0vW01,p(Ω),

so u is a nontrivial weak solution of the problem

Δpu=λu+p1+u+p1in Ωu=0on Ω.

Then u > 0 in Ω and its interior normal derivative ∂u/∂ν > 0 on ∂Ω by the strong maximum principle and the Hopf lemma for the p-Laplacian (see Vázquez [17]). Since uμju in C01(Ω), then uμj > 0 in Ω for all sufficiently large j.

It remains to show that uμ minimizes Iμ on 𝓝μ when it is positive. For each w ∈ 𝓝μ, we will construct a path ywΓ such that

maxuyw([0,1])Iμ(u)=Iμ(w).

Since

Iμ(uμ)=cμmaxuyw([0,1])Iμ(u)

by the definition of cμ, the desired conclusion will then follow. First we note that the function

g(t)=Iμ(tw)=tppΩ(|w|pλwp)dxtppΩwpdx+μtΩwdx,t0

has a unique maximum at t = 1. Indeed,

g(t)=tp1Ω(|w|pλwp)dxtp1Ωwpdx+μΩwdx=tp1tp1Ω(|w|pλwp)dx+1tp1μΩwdx

since w ∈ 𝓝μ, and the last two integrals are positive since λ < λ1 and w > 0, so g′(t) > 0 for 0 ≤ t < 1, g′(1) = 0, and g′(t) < 0 for t > 1. Hence

maxt0Iμ(tw)=Iμ(w)>0

since g(0) = 0. In view of Lemma 3.1 (ii), now it suffices to observe that there exists > max {1, R} such that

Iμ(R~u)=R~ppΩ(|u|pλup)dxR~ppΩupdx+μR~Ωudx0

for all u on the line segment joining w to vε since all norms on a finite dimensional space are equivalent.□

Acknowledgement

The third author was supported by the National Research Foundation of Korea Grant funded by the Korea Government (MEST) (NRF-2015R1D1A3A01019789).

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About the article

Received: 2017-12-26

Accepted: 2018-10-14

Published Online: 2019-06-16

Published in Print: 2019-03-01


Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 516–525, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2020-0012.

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© 2020 K. Perera et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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