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

Kanishka Perera; Ratnasingham Shivaji; Inbo Sim

doi:10.1515/anona-2020-0012

Article Open Access

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

Kanishka Perera , Ratnasingham Shivaji and Inbo Sim

Published/Copyright: June 16, 2019

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From the journal Advances in Nonlinear Analysis Volume 9 Issue 1

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 C^1,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=uq−1−μin Ωu>0in Ωu=0on ∂Ω, (1.1)

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

−Δpu=μ(q−p)/(q−1)uq−1−1,

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=λup−1+up∗−1−μin 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 u ∈ W01,p (Ω) of problem (1.2) is given by

Iμ(u)=∫Ω(|∇u|pp−λupp−up∗p∗+μu)dx,

and clearly all weak solutions lie on the set

Nμ=u∈W01,p(Ω):u>0 in Ω and ∫Ω|∇u|pdx=∫Ωλup+up∗−μudx.

We will refer to a weak solution that minimizes I_μ on 𝓝_μ as a ground state. Let

λ1=infu∈W01,p(Ω)∖0∫Ω|∇u|pdx∫Ω|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 N ≥ p² and λ ∈ (0, λ₁), then there exists μ^∗ > 0 such that for all μ ∈ (0, μ^∗), problem (1.2) has a ground state solution u_μ ∈ C^1,α(Ω) for some α ∈ (0, 1).

The scaling u ↦ μ^{–1/(p^∗–p)} u transforms the first equation in the critical semipositone p-Laplacian problem

−Δpu=λup−1+μup∗−1−1in Ωu>0in Ωu=0on ∂Ω (1.4)

into

−Δpu=λup−1+up∗−1−μ(p∗−1)/(p∗−p),

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

Theorem 1.2

If N ≥ p² and λ ∈ (0, λ₁), then there exists μ^∗ > 0 such that for all μ ∈ (0, μ^∗), problem (1.4) has a ground state solution u_μ ∈ C^1,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+p−1+u+p∗−1−μf(u)in Ωu=0on ∂Ω, (1.5)

where u₊(x) = max {u(x), 0} and

f(t)=1,t≥01−|t|p−1,−1<t<00,t≤−1.

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

Iμ(u)=∫Ω(|∇u|pp−λu+pp−u+p∗p∗)dx+μ[∫u≥0udx+∫−1<u<0u−|u|p−1updx−1−1pu≤−1],u∈W01,p(Ω),

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

c<1NSN/p−1−1pμΩ,

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 u_μ 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 μ_j → 0, a subsequence of u_{μ_j} is positive in Ω. The idea is to show that a subsequence of u_{μ_j} converges in C01 (Ω) to a solution of the limit problem

−Δpu=λup−1+up∗−1in Ωu>0in Ωu=0on ∂Ω.

This requires a uniform C^1,α(Ω) estimate of u_{μ_j} for some α ∈ (0, 1). We will obtain such an estimate by showing that u_{μ_j} is uniformly bounded in W01,p (Ω) and uniformly equi-integrable in L^{p^∗}(Ω), and applying a result of de Figueiredo et al. [10]. The proof of uniform equi-integrability in L^{p^∗}(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=μup−1+up∗−1−1in Ωu>0in Ωu=0on ∂Ω

for small μ remains open.

2 Preliminaries

Let

S=infu∈W01,p(Ω)∖0∫Ω|∇u|pdx∫Ω|u|p∗dxp/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, I_μ satisfies the (PS)_c condition for all

c<1NSN/p−1−1pμΩ. (2.2)

Proof

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

Iμ(uj)=∫Ω(|∇uj|pp−λuj+pp−uj+p∗p∗)dx+μ[∫uj≥0ujdx+∫−1<uj<0uj−|uj|p−1ujpdx−1−1puj≤−1]=c+o(1) (2.3)

and

Iμ′(uj)v=∫Ω|∇uj|p−2∇uj⋅∇v−λuj+p−1v−uj+p∗−1vdx+μ[∫uj≥0vdx+∫−1<uj<01−|uj|p−1vdx]=o(1)v∀v∈W01,p(Ω). (2.4)

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

1N∫Ωuj+p∗dx≤c+1−1pμΩ+o(1)uj+1, (2.5)

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

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

|∇uj+|pdx→w∗κ,uj+p∗dx→w∗ν (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 x_i ∈ Ω, i ∈ I such that

κ≥|∇v|pdx+∑i∈Iκiδxi,ν=vp∗dx+∑i∈Iνiδxi, (2.7)

where κ_i, ν_i > 0 and νip/p∗ ≤ κ_i/S. We claim that I = ∅. Suppose by contradiction that there exists i ∈ I. Let φ : ℝ^N → [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)=φx−xiρ,x∈RN

for i ∈ I and ρ > 0, and note that φ_i,ρ : ℝ^N → [0, 1] is a smooth function such that φ_i,ρ(x) = 1 for |x – x_i| ≤ ρ and φ_i,ρ(x) = 0 for |x – x_i| ≥ 2 ρ. The sequence (φ_i,ρ u_j+) is bounded in W01,p (Ω) and hence taking v = φ_i,ρ u_j+ in (2.4) gives

∫Ω(φi,ρ|∇uj+|p+uj+|∇uj+|p−2∇uj+⋅∇φi,ρ−λφi,ρuj+p−φi,ρuj+p∗+μφi,ρuj+)dx=o(1). (2.8)

By (2.6),

∫Ωφi,ρ|∇uj+|pdx→∫Ωφi,ρdκ,∫Ωφi,ρuj+p∗dx→∫Ωφi,ρdν.

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

∫Ω(uj+|∇uj+|p−2∇uj+⋅∇φi,ρ−λφi,ρuj+p+μφi,ρuj+)dx≤C1ρ+μIj1/p+Ij, >

where

Ij:=∫Ω∩B2ρ(xi)uj+pdx→∫Ω∩B2ρ(xi)vpdx≤Cρp∫Ω∩B2ρ(xi)vp∗dxp/p∗.

So passing to the limit in (2.8) gives

∫Ωφi,ρdκ−∫Ωφi,ρdν≤C(1+μρ)∫Ω∩B2ρ(xi)vp∗dx1/p∗+∫Ω∩B2ρ(xi)vpdx.

Letting ρ ↘ 0 and using (2.7) now gives κ_i ≤ ν_i, which together with ν_i > 0 and νip/p∗ ≤ κ_i/S then gives ν_i ≥ S^N/p. On the other hand, passing to the limit in (2.5) and using (2.6) and (2.7) gives

νi≤Nc+1−1pμΩ<SN/p

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

∫Ωuj+p∗dx→∫Ωvp∗dx. (2.9)

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

|uj+p∗−1(uj−u)|≤uj+p∗+uj+p∗−1|u|≤2−1p∗uj+p∗+1p∗|u|p∗

by Young’s inequality,

∫Ωuj+p∗−1(uj−u)dx→0

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

∫Ω|∇uj|p−2∇uj⋅∇(uj−u)dx→0,

so u_j → u 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(N−p)/p2(ε+|x|p/(p−1))(N−p)/p,ε>0

when Ω = ℝ^N, where the constant C_N,p > 0 is chosen so that

∫RN|∇uε|pdx=∫RNuεp∗dx=SN/p.

Without loss of generality, we may assume that 0 ∈ Ω. Let r > 0 be so small that B_2r(0) ⊂ Ω, take a function ψ ∈ C0∞ (B_2r(0), [0, 1]) such that ψ = 1 on B_r(0), and set

u~ε(x)=ψ(x)uε(x),vε(x)=u~ε(x)∫Ωu~εp∗dx1/p∗,

so that ∫Ωvεp∗dx=1. Then we have the well-known estimates

∫Ω|∇vε|pdx≤S+Cε(N−p)/p, (2.10)

∫Ωvεpdx≥1Cεp−1,N>p21Cεp−1|log⁡ε|,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 I_μ has a uniformly positive mountain pass level below the threshold for compactness given in Lemma 2.1 for all sufficiently small μ > 0. Let v_ε be as in the last section.

Lemma 3.1

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

∥u∥ = ρ ⇒ I_μ(u) ≥ c₀,
I_μ(tv_ε) ≤ 0 for all t ≥ R and ε ∈ (0, 1],
denoting by Γ = {y ∈ C([0, 1], W01,p (Ω)) : y(0) = 0, y(1) = Rv_ε} the class of paths joining the origin to Rv_ε,

c0≤cμ:=infy∈Γmaxu∈y([0,1])Iμ(u)≤β−1−1pμΩ (3.1)

for all sufficiently small ε > 0,
I_μ has a critical point u_μ at the level c_μ.

Proof

By (1.3) and (2.1),

Iμ(u)≥1p1−λλ1up−S−p∗/pp∗up∗−1−1pμΩ,

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

Since v_ε ≥ 0,

Iμ(tvε)=tpp∫Ω(|∇vε|p−λvεp)dx−tp∗p∗+μt∫Ωvεdx

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

μt∫Ωvεdx≤μtΩ1−1/p∫Ωvεpdx1/p≤Cλμp/(p−1)+λtp2p∫Ωvεpdx,

where

Cλ=1−1p2λ1/(p−1)Ω,

Iμ(tvε)≤tpp∫Ω|∇vε|p−λ2vεpdx−tp∗p∗+Cλμp/(p−1). (3.2)

Then by (2.10) and for ε, μ ∈ (0, 1],

Iμ(tvε)≤(S+C)tpp−tp∗p∗+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

cμ≤1N∫Ω|∇vε|p−λ2vεpdxN/p+Cλμp/(p−1),

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 N ≥ p² 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 u_μ is uniformly bounded in W01,p (Ω) and uniformly equi-integrable in L^{p^∗}(Ω), and hence also uniformly bounded in C^1,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_∗):

u_μ is uniformly bounded in W01,p (Ω),
∫E|uμ|p∗dx→0 as E→0, uniformly in μ,
u_μ is uniformly bounded in C^1,α(Ω) for some α ∈ (0, 1).

Proof

We have

Iμ(uμ)=∫Ω(|∇uμ|pp−λuμ+pp−uμ+p∗p∗)dx+μ[∫uμ≥0uμdx+∫−1<uμ<0uμ−|uμ|p−1uμpdx−1−1puμ≤−1]=cμ (3.3)

and

Iμ′(uμ)v=∫Ω|∇uμ|p−2∇uμ⋅∇v−λuμ+p−1v−uμ+p∗−1vdx+μ[∫uμ≥0vdx +∫−1<uμ<01−|uμ|p−1vdx]=0∀v∈W01,p(Ω). (3.4)

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

1N∫Ωuμ+p∗dx≤cμ+1−1pμΩ≤β (3.5)

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

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

lim_⁡∫Ej|uμj|p∗dx>0. (3.6)

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

|∇uμj+|pdx→w∗κ,uμj+p∗dx→w∗ν (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 x_i ∈ Ω, i ∈ I such that

κ≥|∇v|pdx+∑i∈Iκiδxi,ν=vp∗dx+∑i∈Iνiδxi, (3.8)

where κ_i, ν_i > 0 and νip/p∗ ≤ κ_i/S. Suppose I is nonempty, say, i ∈ I. An argument similar to that in the proof of Lemma 2.1 shows that κ_i ≤ ν_i, so ν_i ≥ S^N/p. On the other hand, passing to the limit in (3.5) with μ = μ_j and using (3.7) and (3.8) gives ν_i ≤ Nβ < S^N/p, a contradiction. Hence I = ∅ and

∫Ωuμj+p∗dx→∫Ωvp∗dx.

As in the proof of Lemma 2.1, a further subsequence of (u_{μ_j}) then converges to some u in W01,p (Ω), and hence also in L^{p^∗}(Ω), and a.e. in Ω. Then

∫Ej|uμj|p∗dx≤∫Ω|uμj|p∗−|u|p∗dx+∫Ej|u|p∗dx→0,

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 u_μ 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 μ_j → 0, a subsequence of u_{μ_j} is positive in Ω. By Lemma 3.2 (iii), a renamed subsequence of u_{μ_j} converges to some u in C01 (Ω). We have

Iμj(uμj)=∫Ω(|∇uμj|pp−λuμj+pp−uμj+p∗p∗)dx+μj[∫uμj≥0uμjdx +∫−1<uμj<0uμj−|uμj|p−1uμjpdx−1−1puμj≤−1]=cμj≥c0

by (3.1) and

Iμj′(uμj)v=∫Ω|∇uμj|p−2∇uμj⋅∇v−λuμj+p−1v−uμj+p∗−1vdx+μj[∫uμj≥0vdx +∫−1<uμj<01−|uμj|p−1vdx]=0∀v∈W01,p(Ω),

and passing to the limits gives

∫Ω(|∇u|pp−λu+pp−u+p∗p∗)dx≥c0

and

∫Ω|∇u|p−2∇u⋅∇v−λu+p−1v−u+p∗−1vdx=0∀v∈W01,p(Ω),

so u is a nontrivial weak solution of the problem

−Δpu=λu+p−1+u+p∗−1in 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 u_{μ_j} → u in C01 (Ω), then u_{μ_j} > 0 in Ω for all sufficiently large j.

It remains to show that u_μ minimizes I_μ on 𝓝_μ when it is positive. For each w ∈ 𝓝_μ, we will construct a path y_w ∈ Γ such that

maxu∈yw([0,1])Iμ(u)=Iμ(w).

Since

Iμ(uμ)=cμ≤maxu∈yw([0,1])Iμ(u)

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

g(t)=Iμ(tw)=tpp∫Ω(|∇w|p−λwp)dx−tp∗p∗∫Ωwp∗dx+μt∫Ωwdx,t≥0

has a unique maximum at t = 1. Indeed,

g′(t)=tp−1∫Ω(|∇w|p−λwp)dx−tp∗−1∫Ωwp∗dx+μ∫Ωwdx =tp−1−tp∗−1∫Ω(|∇w|p−λwp)dx+1−tp∗−1μ∫Ωwdx

since w ∈ 𝓝_μ, 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

maxt≥0Iμ(tw)=Iμ(w)>0

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

Iμ(R~u)=R~pp∫Ω(|∇u|p−λup)dx−R~p∗p∗∫Ωup∗dx+μR~∫Ωudx≤0

for all u on the line segment joining w to v_ε 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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Received: 2017-12-26

Accepted: 2018-10-14

Published Online: 2019-06-16

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

Articles in the same Issue

https://doi.org/10.1515/anona-2020-0012

Keywords for this article

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

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