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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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Ground state solutions for a semilinear elliptic problem with critical-subcritical growth

Claudianor O. Alves
  • Corresponding author
  • Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, Campina Grande, PB, 58.109-970, Brazil
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/ Grey Ercole / M. Daniel Huamán Bolaños
Published Online: 2018-08-07 | DOI: https://doi.org/10.1515/anona-2017-0170

Abstract

We prove the existence of at least one ground state solution for the semilinear elliptic problem

{-Δu=up(x)-1,u>0,inGN,N3,uD01,2(G),

where G is either N or a bounded domain, and p:G is a continuous function assuming critical and subcritical values.

Keywords: Variational methods; positive solutions; critical growth

MSC 2010: 35J20; 35J61; 35B33

1 Introduction

In this paper we deal with the existence of a ground state solution for the semilinear elliptic problem

{-Δu=up(x)-1,u>0,inG,uD01,2(G),(P)

where either G=N and D01,2(G)=D1,2(N) or G is a bounded domain in N and D01,2(G)=H01(G). In both cases, N3 and p:G is a continuous function satisfying the following condition:

  • (H1)

    There exist a bounded set ΩG, with positive N-dimensional Lebesgue measure, and positive constants p-, p+ and δ such that

    2<p-p(x)p+<2*for allxΩ,(H1a)p(x)2*for allxGΩδ,(H1b)2<p-p(x)<2*for allxΩδ,(H1c)

    where

    Ωδ:={xG:dist(x,Ω¯)δ}

    and 2*:=2NN-2 is the critical Sobolev exponent.

The energy functional associated with (P) is given by

I(u):=12G|u|2dx-G1p(x)(u+)p(x)dx

and the corresponding Nehari manifold is defined by

𝒩:={uD01,2(G){0}:I(u)(u)=0}.

Our goal in this paper is to obtain a critical point u of I satisfying u>0 in G and

I(u)=infv𝒩I(v).

We refer to such a critical point as a ground state solution of (P).

There are several works in the literature dealing with semilinear problems in the particular case p(x)2*. Let us mention some of them.

In [19], Pohozaev showed that the problem

{-Δu=λu+|u|2*-2u,u>0,inG,uH01(G)(P1)

does not admit a nontrivial solution if λ0, provided that the bounded domain G is strictly star-shaped with respect to the origin in N, N3.

In [7], Brezis and Nirenberg proved the following results: if N4, problem (P1) has a positive solution for every λ(0,λ1), where λ1 denotes the first eigenvalue of (-Δ,H01(Ω)); if N=3, there exists λ*[0,λ1) such that for any λ(λ*,λ1) problem (P1) admits a positive solution. In the particular case where G is a ball, they proved that a positive solution exists if, and only if, λ(λ1/4,λ1) and also that when N=3 there exists λ*>0 such that (P1) does not have a solution for λλ*.

In [9], Coron proved that if there exist R,r>0 such that

G{xN:r<|x|<R}andG¯{xN:|x|<r}

and the ratio R/r is sufficiently large, then problem (P1) with λ=0 has a positive solution in H01(G).

In [6], Bahri and Coron showed that if λ=0 and i(G;/2)0 (ith homology group) for some i>0, then problem (P1) has at least one positive solution. (When i(G;/2)0 the boundary G is not connected.)

Existence results for (P1) related to the topology of G were also obtained by Bahri, in [5]. In [8], Carpio, Comte and Lewandowski obtained nonexistence results for (P1), with λ=0, in contractible non-star-shaped domains.

On the other hand, the subcritical problem

{-Δu=|u|q-2uinG,2<q<2*,u=0onG(P2)

has an unbounded set of solutions in H01(G) (see [13]).

Problem (P2) with q=2*-ϵ (ϵ>0) was studied in the papers [3] and [14]. In the former, Atkinson and Peletier considered G a ball and determined the exact asymptotic behavior of the corresponding (radial) solutions uϵ, as ϵ0. In [14], where a general bounded domain G was considered, Garcia Azorero and Peral Alonso provided an alternative for the asymptotic behavior of Jϵ(uϵ), as ϵ0, where Jϵ denotes the energy functional associated with problem (P2) and q=2*-ϵ. More precisely, they showed that if

1NSN2<limϵ0Jϵ(uϵ)<2NSN2,

where S denotes the Sobolev constant, then uϵ converges to either a Dirac mass or a solution of the critical problem

{-Δu=|u|2*-2uinG,u=0onG.

We recall that the Sobolev constant is defined by

S:=inf{u1,22u2*2:uD1,2(N){0}}(1.1)

and given explicitly by the expression

S:=πN(N-2)(Γ(N/2)Γ(N))2N,

where Γ(t)=0st-1e-sds is the Gamma function (see the papers [4] and [20] by Aubin and Talenti, respectively). Furthermore, S is achieved in (1.1) by the Aubin–Talenti function

w(x)=[N(N-2)](N-2)/4(1+|x|2)(n-2)/2(1.2)

(and also by translations, rescalings and scalar multiples of it). This function also satisfies

N|w|2dx=N|w|2*dx=SN2

and is a ground state solution of problem (P) with G=N and p(x)2.

In [16], Kurata and Shioji studied the compactness of the embedding H01(G)Lp(x)(G) for a bounded domain G and a variable exponent 1p(x)2*. (For the definition and properties of Lp(x)(G) see [12]). They showed the existence of a positive solution of (P) under the hypothesis of existence of a point x0G, a small η>0, 0<l<1 and c0>0 such that p(x0)=2* and

p(x)2*-c0(log(1/|x-x0|))l,|x-x0|η.

In [1], Alves and Souto studied the existence of nonnegative solutions for the equation

-div(|u|p(x)-2u)=uq(x)-1inN,

where the variable exponents p(x) and q(x) are radially symmetric functions satisfying 1<essinfNp(x)esssupNp(x)<N, p(x)q(x)2* and

p(x)=2,q(x)=2*if either|x|δor|x|R,

for constants 0<δ<R.

Finally, in [18], Liu, Liao and Tang proved the existence of a ground state solution for (P) with G=N and

p(x)={pifxΩ,2*ifxNΩ,

where the constant p belongs to (2,2*) and ΩN has nonempty interior.

In Section 2, motivated by the results of [18], we use the concentration-compactness lemma by P. L. Lions and properties of the Nehari manifold 𝒩 to prove the existence of at least one ground state for problem (P) when G=N and pC(N,) is a function satisfying condition (H1). A key point in the proof of our existence result is the achievement of the strict inequality

infv𝒩I(v)<SN2N(1.3)

and we get this by exploring the “projection” on the Nehari manifold of the sequence (wk), where wk(x)=w(x+keN) and eN=(0,0,,1) is the Nth coordinate vector.

In Section 3, we study the case where G is a bounded domain in N. In this case, the argument based on the sequences of translations of the Aubin–Talenti function is not applicable. Thus, in order to achieve the inequality (1.3) we assume an additional hypothesis (H2) that is stated in terms of a subdomain U of Ω and the value

q¯:=min{q(2,2*]:g(q)=1NSN2},(1.4)

where the function g:(2,2*](0,) is given by

g(q)=(12-1q)Sq(U)qq-2

and Sq(U) denotes the best constant of the embedding H01(U)Lq(U), that is,

Sq(U):=inf{uL2(U)2uLq(U)2:uH01(U){0}}.

More precisely, we assume that the function pC(G,), satisfying (H1), also verifies the following hypothesis, where Ω, p- and p+ are defined in (H1) and q¯ is defined by (1.4):

  • (H2)

    There exists a subdomain U of Ω such that

    S2(U)1andp-p(x)q<min{q¯,p+}for allxU.

Under the hypotheses (H1) and (H2), we show that problem (P) has at least one ground state solution.

We remark that the constant q in the statement of (H2) can be any lower bound for q¯ in the interval (p-,p+). Considering that the particular value S2(U) is available for several domains (especially those with some kind of symmetry), we derive two lower bounds q1>q2 for q¯ in terms of S2(U), S and |U| and a third, q3, depending only on S and |U|. Moreover, we present sufficient conditions for S2(U)1 to hold, when the subdomain U is either a ball BR or an annular-shaped domain BRBr¯, with Br¯BR. We also show that if R and R-r are sufficiently large, then S2(U)<1 for U=BR and U=BRBr¯, respectively.

2 The semilinear elliptic problem in N

In this section, we consider the semilinear elliptic problem with variable exponent

{-Δu=up(x)-1,u>0,inN,uD1,2(N),(2.1)

where N3 and p:N is a continuous function verifying hypothesis (H1).

We recall that the space D1,2(N) is the completion of C0(N) with respect to the norm

u1,2:=(N|u|2dx)12.

The dual space of D1,2(N) will be denoted by D-1.

The energy functional I:D1,2(N) associated with (2.1) is given by

I(u)=12u1,22-N1p(x)(u+)p(x)dx,

where u+(x)=max{u(x),0}. Hence, under hypothesis (H1), we can write

I(u)=12u1,22-Ωδ1p(x)(u+)p(x)dx-12*NΩδ(u+)2*dx.

For a posterior use, let us estimate the second term in the above expression. For this, let uD1,2(N) and consider the set E={xΩδ:|u(x)|<1}. Then

Ωδ1p(x)(u+)p(x)dx1p-E(u+)p-dx+1p-ΩδE(u+)2*dx1p-Ωδ|u|p-dx+1p-u2*2*1p-(Ωδ|u|2*dx)p-2*|Ωδ|2*-p-2*+1p-u2*2*1p-|Ωδ|2-p-2*u2*p-+1p-u2*2*,

where we have used (H1) and Hölder’s inequality. Hence, it follows from (1.1) and (H1c) that

12*Ωδ(u+)p(x)dxΩδ1p(x)(u+)p(x)dxau1,2p-+bu1,22*,(2.2)

where

a=1p-|Ωδ|2*-p-2*S-p-2andb=S-2*2p-.

We observe from (2.2) that the functional I is well defined.

The next lemma establishes that I is of class C1. Since its proof is standard, it will be omitted.

Lemma 2.1.

Let pC(RN,R) a function satisfying (H1a). Then IC1(D1,2(RN),R) and

I(u)(v)=Nuvdx-N(u+)p(x)-1vdxfor allu,vD1,2(N).

Remark 2.2.

The previous lemma ensures that uD1,2(N) is a weak solution of (2.1) if, and only if, u is a critical point of I (i.e. I(u)=0). We remark that a critical point u of I is nonnegative, since

0=I(u)(u-)=Nuu-dx-N(u+)p(x)-1u-dx=u-1,22,

where u-(x)=min{u(x),0}. Consequently, according to the Strong Maximum Principle, if u0 is a critical point of I, then u>0 in N.

2.1 The Nehari manifold

In this subsection we prove some properties of the Nehari manifold associated with (2.1), which is defined by

𝒩:={uD1,2(N){0}:J(u)=0},

where

J(u):=I(u)(u)=u1,22-N(u+)p(x)dx.

Of course, critical points of I belong to 𝒩.

Definition 2.3.

We say that u𝒩 is a ground state solution for (2.1) if I(u)=0 and I(u)=m, where

m:=infu𝒩I(u).

In the sequel we show important properties involving the Nehari manifold, which are crucial in our approach.

Proposition 2.4.

Assume that (H1) holds. Then m>0.

Proof.

For an arbitrary u𝒩 we have

u1,22=N(u+)p(x)dx=Ωδ(u+)p(x)dx+NΩδ(u+)p(x)dxΩδ(u+)p(x)dx+S-2*2u1,22*.

Thus, it follows from (2.2) that

u1,22C1u1,2p-+C2u1,22*,

where C1 and C2 denote positive constants that do not depend on u. Consequently,

1C1u1,2p--2+C2u1,22*-2,

from which we conclude that there exists η>0 such that

u1,2ηfor allu𝒩.(2.3)

Therefore,

I(u)=I(u)-1p-I(u)(u)=(12-1p-)u1,22+N(1p--1p(x))(u+)p(x)dx(12-1p-)u1,22.(2.4)

In view of (2.3), this implies that m(12-1p-)η2>0. ∎

Proposition 2.5.

Assume (H1). Then, for each uD1,2(RN) with u+0, there exists a unique tu>0 such that tuuN.

Proof.

Let

f(t):=I(tu)=t22u1,22-Ntp(x)p(x)(u+)p(x)dx,t(0,+).

We note that

f(t)=I(tu)(u)=tu1,22-Ntp(x)-1(u+)p(x)dx=1tJ(tu)for allt(0,+).

Since 1<p--1p(x)-1, we have

f(t)t(u1,22-tp--2N(u+)p(x)dx)for allt(0,1),f(t)t(u1,22-tp--2N(u+)p(x)dx)for allt1.

Thus, we can see that f(t)>0 for all t>0 sufficiently small and also that f(t)<0 for all t1 sufficiently large. Therefore, there exists tu>0 such that

f(tu)=1tuJ(tuu)=0,

showing that tuu𝒩.

In order to prove the uniqueness of tu, let us assume that 0<t1<t2 satisfy f(t1)=f(t2)=0. Then

u1,22=Nt1p(x)-2(u+)p(x)dx=Nt2p(x)-2(u+)p(x)dx.

Hence,

N(t1p(x)-2-t2p(x)-2)(u+)p(x)dx=0.

Since t1p(x)-2<t2p(x)-2 for all xN, the above equality leads to the contradiction u+0. ∎

Proposition 2.6.

Assume that (H1) holds. Then

J(u)(u)(2-p-)η2<0for allu𝒩,

where η was given in (2.3). Hence, J(u)0 for all uN.

Proof.

For u𝒩 we have

J(u)(u)=2u1,22-Np(x)(u+)p(x)dx2u1,22-p-N(u+)p(x)dx=(2-p-)u1,22(2-p-)η2<0,

according to (2.3). ∎

Proposition 2.7.

Assume (H1) and that there exists u0N such that I(u0)=m. Then u0 is ground state solution for (2.1) and u0>0 in RN.

Proof.

Since m is the minimum of I on 𝒩, Lagrange multiplier theorem implies that there exists λ such that I(u0)=λJ(u0). Thus

λJ(u0)(u0)=I(u0)(u0)=J(u0)=0.

According to the previous proposition, λ=0, and so, I(u0)=0.

Proposition 2.4 implies that u00. Therefore, u0>0 in N (see Remark 2.2). ∎

The next proposition shows that, under (H1), there exists a Palais–Smale sequence for I associated with the minimum m.

Proposition 2.8.

Assume (H1). There exists a sequence (un)N such that: un0 in RN, I(un)m and I(un)0 in D-1.

Proof.

According to the Ekeland variational principle (see [21, Theorem 8.5]), there exist (vn)𝒩 and (λn) such that

I(vn)mandI(vn)-λnJ(vn)0inD-1.

It follows from (2.4) that

(12-1p-)vn1,22I(vn).

This implies that (vn) is bounded in D1,2(N). Hence, taking into account that

|I(vn)(vn)-λnJ(vn)(vn)|I(vn)-λnJ(vn)D-1vn1,2,

we have

I(vn)(vn)-λnJ(vn)(vn)0.

Using the fact that I(vn)(vn)=0, we conclude from Proposition 2.6 that λn0. Consequently, I(vn)0 in D-1.

We affirm that the sequence (vn+) satisfies I(vn+)m and I(vn+)0 in D-1. Indeed, since

vn-1,2=I(vn)(vn-)0,

we derive

I(vn+)=I(vn)-12vn-1,22m.

Moreover,

|I(vn+)|D-1=sup|ϕ|1I(vn+)(ϕ)=sup|ϕ|1I(vn)(ϕ)-N(vn-)ϕdxI(vn)D-1+vn-1,20.

Now, let us fix tn>0 such that un:=tnvn+𝒩. Using the fact that I(un)un=0 and I(vn+)vn+=on(1), a simple computation gives

tn1,

so that

I(un)-I(vn+)=on(1)andI(un)-I(vn+)=on(1).

Hence,

un𝒩,un0,I(un)mandI(un)0.

This proves the proposition. ∎

The next proposition provides a special upper bound for m.

Proposition 2.9.

Assume (H1). Then m<1NSN2, where S denotes the Sobolev constant defined by (1.1).

Proof.

Let

wk(x):=w(x+keN),eN=(0,0,,0,1),

where w:N is the Aubin–Talenti function given by (1.2), which satisfies

w1,22=w2*2*=SN2.

A direct computation shows that wk2*=w2* and wk1,2=w1,2. Moreover, exploring the expression of w, we can easily check that wk0 uniformly in bounded sets and, therefore,

limkΩδ|wk|αdx=0(2.5)

for any α>0.

By Proposition 2.5, there exists tk>0 such that tkwk𝒩, which means that

tk2wk1,22=Ωδ(tkwk)p(x)dx+NΩδ(tkwk)2*dx.

Hence,

w1,22=wk1,22tk2*-2NΩδ|wk|2*dx

and then, by using (2.5) for α=2*, we can verify that the sequence (tk) is bounded,

lim supktklim supk(w1,22NΩδ(wk)2*dx)12*-2=(w1,22w2*2*)12*-2=1.

Moreover, since tkwk𝒩,

mI(tkwk)=tk22wk1,22-NΩδ(tkwk)2*2*dx-Ωδ(tkwk)p(x)p(x)dx=tk22SN2-N(tkwk)2*2*dx+Ωδ(tkwk)2*2*dx-Ωδ(tkwk)p(x)p(x)dx=SN2(tk22-tk2*2*)+Ωδ((tkwk)2*2*-(tkwk)p(x)p(x))dxSN2N+Ωδ((tkwk)2*2*-(tkwk)p(x)p(x))dx,

where we have used that the maximum value of the function t[0,)t22-t2*2* is 1N.

Combining the boundedness of the sequence (tk) with the fact that wk0 uniformly in Ωδ, we can select k sufficiently large, such that tkwk1 in Ωδ. Therefore, for this k,

mSN2N+Ωδ((tkwk)2*2*-(tkwk)2*p(x))dx=SN2N+tk2*Ωδ(wk)2*(12*-1p(x))dx<SN2N,

since the latter integrand is strictly negative in Ω, which has positive N-dimensional Lebesgue measure. ∎

2.2 Existence of a ground state solution

Our main result in this section is the following.

Theorem 2.10.

Assume that (H1) holds. Then problem (2.1) has at least one ground state solution.

We prove this theorem throughout this subsection by using the following well-known result.

Lemma 2.11 (Lions’ lemma [17]).

Let (un) be a sequence in D1,2(RN), N>2, satisfying

  • unu𝑖𝑛D1,2(N),

  • |un|2μ𝑖𝑛(N),

  • |un|2*ν𝑖𝑛(N).

Then there exist an at most countable set of indices I, points (xi)iI and positive numbers (νi)iI such that

  • (i)

    ν=|u|2*+iνiδxi,

  • (ii)

    μ({xi})νi2/2*S for any i,

where δxi denotes the Dirac measure supported at xi.

We know from Proposition 2.8 that there exists a sequence (un)𝒩 satisfying un0 in N, I(un)m and I(un)0 in D-1. Since (un) is bounded in D1,2(N), we can assume (by passing to a subsequence) that there exists uD1,2(N) such that unu in D1,2(N), unu in Llocs(N) for 1s<2* and un(x)u(x) a.e. in N. Moreover, |un|2μ and |un|2*ν in (N).

We claim that u0. Indeed, let us suppose, by contradiction, that u0. We affirm that this assumption implies that the set given by Lions’ lemma is empty. Otherwise, let us fix i, xiN and νi>0 as in Lions’ lemma. Let ϕCc(N) such that

ϕ(x)={1,xB1(0),0,xB2(0),

and 0ϕ(x)1 for all xN, where B1 and B2 denotes the balls centered at the origin, with radius 1 and 2, respectively.

For ϵ>0 fixed, define

ϕϵ(x)=ϕ(x-xiϵ).

Since (un) is bounded in D1,2(N), the same holds for the sequence (ϕϵun). Thus,

|I(un)(ϕϵun)|I(un)D-1ϕϵun1,2=on(1),

so that

Nun(ϕϵun)dx=N(un)p(x)ϕϵdx+on(1).

Consequently,

Nϕϵ|un|2dx+NununϕϵdxN|un|p-ϕϵdx+N|un|2*ϕϵdx+on(1).(2.6)

According to Lions’ lemma,

N|un|2ϕϵdxNϕϵdμandN|un|2*ϕϵdxNϕϵdν.

Since

|Nununϕϵdx|ϕϵ(B2ϵ(xi)|un|2dx)12un1,20

and

N|un|p-ϕϵdx0,

it follows from (2.6) that

NϕϵdμNϕϵdνfor allϵ>0.

Now, making ϵ0, we get

μ({xi})νi.

Combining this inequality with part (ii) of Lions’ lemma, we obtain νiSN2. It follows that

SN2Sνi2/2*μ({xi})νi.

Let ϕCc(N) such that ϕ(xi)=1 and 0ϕ(x)1, for any xN. Recalling that

I(un)=I(un)-12*I(un)(un)=1Nun1,22+Ωδ(12*-1p(x))|un|p(x)dx,

we have

I(un)1NN|un|2ϕdx+Ωδ(12*-1p(x))|un|p(x)dx.(2.7)

Since p: is continuous, for each ϵ>0, there exists Ωδ,ϵΩδ such that

|12*-1p(x)|<ϵ2M,xΩδΩδ,ϵ,

where M=supn(Ωδ|un|p-+|un|2*dx). Thus,

|Ωδ(12*-1p(x))|un|p(x)dx|ϵ2MΩδΩδ,ϵ|un|p(x)dx+(1p--12*)Ωδ,ϵ|un|p(x)dxϵ2MΩδ(|un|p-+|un|2*)dx+(1p--12*)Ωδ,ϵ(|un|p-+|un|q)dxϵ2+(1p--12*)Ωδ(|un|p-+|un|q)dx,

where 2<p-p(x)q<2*, for xΩδ,ϵ. Then, since un0 in Llocs(N), for s[1,2*), and ϵ is arbitrary, we conclude that

limnΩδ(12*-1p(x))|un|p(x)dx=0.

Therefore, by making n in (2.7), we obtain

m1NNϕdμ1N{xi}ϕdμ=1Nμ({xi})1NSN2

which contradicts Proposition 2.9, showing that =. Hence, it follows from Lions’ lemma that

un0inLloc2*(N).

In particular, un0 in L2*(Ωδ), so that

0Ωδ|un|p(x)dxΩδ|un|p-dx+Ωδ|un|2*dx0.

Since (un)𝒩, we have

limnun1,22=limn(Ωδ|un|p(x)dx+NΩδ|un|2*dx)=limnNΩδ|un|2*dx=:L.

Thus, by making n in the equality

I(un)=12un1,22-Ωδ|un|p(x)p(x)dx-12*NΩδ|un|2*dx,

we obtain

m=12L-12*L=1NL.

Since

Sun1,22un2*2un1,22(NΩδ|un|2*dx)2/2*,

we obtain m=LN1NSN2, which contradicts Proposition 2.9 and proves that u0.

Now, combining the weak convergence

unuinD1,2(N)

with the fact that I(un)0 in D-1, we conclude that

I(u)(v)=0for allvD1,2(N),

meaning that u is a nontrivial critical point of I.

Thus, taking into account Proposition 2.7, in order to complete the proof that u is a ground state solution for (2.1) we need to verify that I(u)=m. Indeed, since

I(un)=(12-1p-)un1,22+N(1p--1p(x))unp(x)dx,

the weak convergence unu in D1,2(N) and Fatou’s lemma imply that

m(12-1p-)lim infnun1,2+lim infnN(1p--1p(x))unp(x)dx(12-1p-)u1,2+N(1p--1p(x))up(x)dx=I(u)-1p-I(u)(u)=I(u)m,

showing that I(u)=m.

3 The semilinear elliptic problem in a bounded domain

In this section we consider the elliptic problem

{-Δu=up(x)-1,u>0,inGu=0onG,(3.1)

where G is a smooth bounded domain of N, N3, and p:G is a continuous function verifying (H1) and an additional hypothesis (H2), which is stated in the sequel.

We recall that the usual norm in H01(G) is given by

u:=u2=(G|u|2dx)12.

We denote the dual space of H01(G) by H-1.

The energy functional I:H01(G) associated with problem (3.1) is defined by

I(u):=12G|u|2dx-G1p(x)(u+)p(x)dx.

It belongs to C1(H01(G),) and its derivative is given by

I(u)(v)=Guvdx-G(u+)p(x)-1vdxfor allu,vH01(G).

Thus, a function uH01(G) is a weak solution of (3.1) if, and only if, u is a critical point of I. Moreover, as in Section 2, the nontrivial critical points of I are positive in G (a consequence of the Strong Maximum Principle).

We maintain the notation of Section 2. Thus,

J(u):=I(u)(u)=u2-G(u+)p(x)dx,

the Nehari manifold associated with (3.1) is defined by

𝒩:={uH01(G){0}:J(u)=0}

and m:=infu𝒩I(u).

Definition 3.1.

We say that u𝒩 is a ground state solution for (3.1) if I(u)=0 and I(u)=m.

We gather in the next lemma some results that can be proved as in Section 2.

Lemma 3.2.

Assume (H1). We claim that:

  • (i)

    m>0.

  • (ii)

    J(u)(u)<0 for all u𝒩 . (Thus, J(u)0 for all u𝒩 .)

  • (iii)

    If I(u0)=m , then I(u0)=0 . (Thus, u0 is a positive weak solution of ( 3.1 ).)

  • (iv)

    There exists a sequence (un)𝒩 such that un0 in G, I(un)m and I(un)0 in H-1.

Unfortunately, hypothesis (H1) by itself is not sufficient to guarantee that m<SN2N as in Proposition 2.9. The reason is that the translation argument used in the proof of that Proposition does not apply to a bounded domain. So we assume an additional assumption (H2). In order to properly state such an assumption, we need some background information.

Let UN be a bounded domain and define

Sq(U):=inf{vL2(U)2vLq(U)2:vH01(U){0}},1q2*.(3.2)

It is well known that if 1q<2*, then the infimum in (3.2) is attained by a positive function ϕq in H01(U). Actually, this follows from the compactness of the embedding H01(U)Lq(U).

Another well-known fact is that in the case q=2* the infimum in (3.2) coincides with the best Sobolev constant, i.e.

S2*(U)=S:=inf{vL2(N)2vL2*(N)2:vD1,2(N){0}}.(3.3)

Moreover, in this case the infimum (3.2) is not attained if U is a proper subset of N.

In the sequence we make use of the function

g(q):=(12-1q)Sq(U)qq-2,q(2,2*].

Lemma 3.3.

If S2(U)1, then there exists q¯(2,2*] such that

g(q)<g(q¯)=1NSN2for allq(2,q¯).(3.4)

Proof.

The following facts are known, where |U| denotes the volume of U (see [2, 10]): the function

q[1,2*]Sq(U)

is continuous (in fact it is α-Hölder continuous, for any 0<α<1, as proved in [11]) and the function

q[1,2*]|U|2qSq(U)

is strictly decreasing. It follows that

limq2+Sq(U)=S2(U)

and

|U|2qSq(U)<|U|S2(U),q(2,2*].

This latter inequality implies that

g(q)<g1(q):=|U|(12-1q)S2(U)qq-2,q(2,2*].(3.5)

Hence, using that S2(U)1, we see that

limq2+g(q)=0.

Taking into account that S2*(U)=S, we can easily check that g(2*)=1NSN2. Thus, defining

q¯:=min{q(2,2*]:g(q)=1NSN2},(3.6)

we arrive at (3.4). ∎

The additional hypothesis (H2) is stated as follows, where Ω, p- and p+ are defined in (H1) and q¯ is given by (3.6):

  • (H2)

    There exists a subdomain U of Ω such that

    S2(U)1andp-p(x)q<min{q¯,p+}for allxU.

Lemma 3.4.

Assume that (H1) and (H2) hold. Then m<1NSN2.

Proof.

Let ϕqH01(U) denote a positive extremal function of Sq(U). Thus, ϕq>0 in U and

Sq(U)=ϕqL2(U)2ϕqLq(U)2.

Let us define the function ϕ~qH01(G) by

ϕ~q(x):={ϕq(x)ifxU,0ifxGU.

For each t>0 we have

I(tϕ~q)=t22G|ϕ~q|2dx-Gtp(x)p(x)(ϕ~q)p(x)dx=α2t2-βqtq,

where

α:=U|ϕq|2dxandβ:=U(ϕq)qdx.

Taking

tq:=(αβ)1q-2,

it is easy to see that tqϕ~𝒩 and

I(tqϕ~q)=(12-1q)(αqβ2)1q-2=(12-1q)Sq(U)qq-2.

Since S2(U)1 and p-q<min{p+,q¯}, it follows from Lemma 3.3 that

g(q)=I(tqϕ~q)<1NSN2.

This implies that m<1NSN2. ∎

The main result in this section is the following.

Theorem 3.5.

Assume (H1) and (H2). Then problem (3.1) has at least one ground state solution.

Proof.

According to item (iv) of Lemma 3.2, there exists a sequence (un)𝒩 satisfying I(un)m and I(un)0 in H-1. Since (un) is bounded in H01(G), there exist uH01(G) and a subsequence, still denoted by (un), such that unu in H01(G), unu in Lp(G), for 1p<2*, and un(x)u(x) a.e. in G. Arguing as in Section 2, we can combine Lions’ lemma and Lemma 3.4 to prove that u0, I(u)=0 and I(u)=m, showing thus that u is a ground state solution of (3.1). ∎

3.1 On hypothesis (H2)

In this subsection we present some lower bounds for the value of q¯, defined by (3.6), which can be used as the value constant for p(x) in hypothesis (H2). Moreover, we give some examples of simple bounded domains U such that S2(U)1.

The value of q¯ depends on the function qg(q), which in turn depends on the function qSq(U). It is well known that Sq(U) is the least value of λ for which the Dirichlet problem

{-Δu=λuL2(U)2-q|u|q-2uinU,u=0onU

has a nontrivial weak solution. When p=2, this is the well-studied eigenvalue problem for (-Δ,H01(U)) and S2(U) is its first eigenvalue. It follows that S2(U) can be found analytically for some simple domains as balls, rectangles and other domains enjoying some kind of symmetry. For instance, if U is a ball of radius R, then

S2(U)=(jN2-1,1R)2,(3.7)

where jα,1 denotes the first positive root of the first kind Bessel function of order α.

When q2 the above problem is no longer linear and, consequently, it is more difficult to be solved analytically, even for simple domains. For this reason, determining an analytical expression for the function g on the interval (2,2*) is a hard task and we do not know the exact value of q¯ given by (3.6). However, the inequality (3.5) allows us to derive lower bounds for q¯, in terms of S2(U), |U| and S, which can be used as the value constant for p(x) in hypothesis (H2). In fact, assuming S2(U)1, we can easily verify that g1(q)>0 for all q(2,2*], where the function qg1(q) is defined in (3.5). Therefore, taking into account that limq2+g1(2)=0 and g1(q¯)>g(q¯)=1NSN2, there exists a unique value q1(2,q¯) such that g1(q1)=1NSN2, that is

|U|(12-1q1)S2(U)q1q1-2=1NSN2,

an equation that can be solved at least numerically.

A rougher but explicit lower bound q2 for q¯ follows from the inequality

g1(q)<g2(q):=|U|NS2(U)qq-2,q(2,2*),

which is obtained from (3.5) by observing that 12-1q12-12*=1N. Indeed, since the function g2 enjoys the same properties as g1, there exists a unique point q2(2,q¯) satisfying g2(q2)=1NSN2. A simple calculation yields

q2:=2log(|U|SN/2)log(S2(U)|U|SN/2).

Of course, 2<q2<q1<q¯.

A third lower bound q3 for q¯ also follows from (3.5). Indeed, by using that S2(U)1 in (3.5), we obtain

g(q)<g3(q):=|U|(12-1q),q(2,2*].

Hence, since g3>0 and g3(2)=0, there exists a unique point q3(2,q¯) satisfying g3(q3)=SN2N. Such a point is given explicitly by

q3:=2N|U|N|U|-2SN2.

Another conclusion that follows easily from the monotonicity of the function q|U|2qSq(U) combined with (3.3) is that if S2(U)1, then |U|>SN2.

In the sequel, we present sufficient conditions for the inequality S2(U)1 to hold when U is either a ball or an annulus. We will denote by BR(y) the ball centered at y with radius R>0. When y=0, we will write simply BR.

Example 3.6.

Let U=BR(y)Ω. Since the Laplacian operator is invariant under translations,

S2(BR)=S2(BR(y)).

Moreover, a simple scaling argument (or (3.7)) yields

S2(BR)=R-2S2(B1).

So, if RS2(B1)12, then S2(U)=S2(BR(y))1.

Example 3.7.

Let U=BR(y)Br(z)¯Ω, with Br(z)¯BR(y), for some y,zΩ and R>r>0. Since the Laplacian operator is invariant under orthogonal transformations, we can see that

S2(BRBr(se1)¯)=S2(U)

for some s[0,R-r), where e1 denotes the first coordinate vector. According to [15, Proposition 3.2], the function tS2(BRBr(te1)¯) is strictly decreasing for t[0,R-r). Therefore,

S2(BRBr¯)S2(U).(3.8)

Since B(R-r)/2 is the largest ball contained in BRBr¯, we have

S2(BRBr¯)<S2(B(R-r)/2)=(R-r2)-2S2(B1).(3.9)

Hence, if R-r2S2(B1)12, then (3.8) and (3.9) imply that S2(U)<1.

Thus, we can replace the condition S(U)1 in (H2) by either RS2(B1)12 when U=BR(y) or R-r2S2(B1)12 when U=BR(y)Br(z)¯.

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About the article

Received: 2017-07-26

Revised: 2018-06-18

Accepted: 2018-06-25

Published Online: 2018-08-07

Published in Print: 2019-03-01


Funding Source: Conselho Nacional de Desenvolvimento Científico e Tecnológico

Award identifier / Grant number: 304036/2013-7

Funding Source: Conselho Nacional de Desenvolvimento Científico e Tecnológico

Award identifier / Grant number: 483970/2013-1

Award identifier / Grant number: 306590/2014-0

Funding Source: Fundação de Amparo à Pesquisa do Estado de Minas Gerais

Award identifier / Grant number: APQ-03372-16

C. O. Alves was partially supported by CNPq/Brazil (304036/2013-7) and INCT-MAT. G. Ercole was partially supported by CNPq/Brazil (483970/2013-1 and 306590/2014-0) and Fapemig/Brazil (APQ-03372-16).


Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 108–123, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0170.

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