Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advances in Nonlinear Analysis

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

IMPACT FACTOR 2018: 6.636

CiteScore 2018: 5.03

SCImago Journal Rank (SJR) 2018: 3.215
Source Normalized Impact per Paper (SNIP) 2018: 3.225

Mathematical Citation Quotient (MCQ) 2018: 3.18

Open Access
See all formats and pricing
More options …

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
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Grey Ercole / M. Daniel Huamán Bolaños
Published Online: 2018-08-07 | DOI: https://doi.org/10.1515/anona-2017-0170


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


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


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)



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

The energy functional associated with (P) is given by


and the corresponding Nehari manifold is defined by


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


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


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


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


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


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


We recall that the Sobolev constant is defined by


and given explicitly by the expression


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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




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


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




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


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.


For an arbitrary u𝒩 we have


Thus, it follows from (2.2) that


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


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

u1,2ηfor allu𝒩.(2.3)



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.




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


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




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.


For u𝒩 we have


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.


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


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.


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


It follows from (2.4) that


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


we have


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


we derive




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


so that




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).




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


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,


for any α>0.

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




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,


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,


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)


  • (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


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


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


so that




According to Lions’ lemma,






it follows from (2.6) that

NϕϵdμNϕϵdνfor allϵ>0.

Now, making ϵ0, we get


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


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


we have


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


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


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


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


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


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


Since (un)𝒩, we have


Thus, by making n in the equality


we obtain




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

Now, combining the weak convergence


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


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


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


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


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,


the Nehari manifold associated with (3.1) is defined by


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)


  • (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


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.


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


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)


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


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


is strictly decreasing. It follows that




This latter inequality implies that


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


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


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.


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


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


For each t>0 we have






it is easy to see that tqϕ~𝒩 and


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


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.


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


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


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


an equation that can be solved at least numerically.

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


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


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


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


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,


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


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


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,


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


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)¯.


  • [1]

    C. O. Alves and M. A. S. Souto, Existence of solutions for a class of problems in N involving the p(x)-Laplacian, Contributions to Nonlinear Analysis, Progr. Nonlinear Differential Equations Appl. 66, Birkhäuser, Basel (2006) 17–32.  Google Scholar

  • [2]

    G. Anello, F. Faraci and A. Iannizzotto, On a problem of Huang concerning best constants in Sobolev embeddings, Ann. Mat. Pura Appl. (4) 194 (2015), no. 3, 767–779.  CrossrefWeb of ScienceGoogle Scholar

  • [3]

    F. V. Atkinson and L. A. Peletier, Elliptic equations with nearly critical growth, J. Differential Equations 70 (1987), no. 3, 349–365.  CrossrefGoogle Scholar

  • [4]

    T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geom. 11 (1976), no. 4, 573–598.  CrossrefGoogle Scholar

  • [5]

    A. Bahri, Critical Points at Infinity in some Variational Problems, Pitman Res. Notes Math. Ser. 182, Longman Scientific & Technical, Harlow, 1989.  Google Scholar

  • [6]

    A. Bahri and J.-M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: The effect of the topology of the domain, Comm. Pure Appl. Math. 41 (1988), no. 3, 253–294.  CrossrefGoogle Scholar

  • [7]

    H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), no. 4, 437–477.  CrossrefGoogle Scholar

  • [8]

    A. Carpio Rodríguez, M. Comte and R. Lewandowski, A nonexistence result for a nonlinear equation involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire 9 (1992), no. 3, 243–261.  CrossrefGoogle Scholar

  • [9]

    J.-M. Coron, Topologie et cas limite des injections de Sobolev, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), no. 7, 209–212.  Google Scholar

  • [10]

    G. Ercole, Absolute continuity of the best Sobolev constant, J. Math. Anal. Appl. 404 (2013), no. 2, 420–428.  CrossrefWeb of ScienceGoogle Scholar

  • [11]

    G. Ercole, Regularity results for the best-Sobolev-constant function, Ann. Mat. Pura Appl. (4) 194 (2015), no. 5, 1381–1392.  CrossrefWeb of ScienceGoogle Scholar

  • [12]

    X. Fan and D. Zhao, On the spaces Lp(x)(Ω) and Wm,p(x)(Ω), J. Math. Anal. Appl. 263 (2001), no. 2, 424–446.  Web of ScienceGoogle Scholar

  • [13]

    J. P. García Azorero and I. Peral Alonso, Existence and nonuniqueness for the p-Laplacian: Nonlinear eigenvalues, Comm. Partial Differential Equations 12 (1987), no. 12, 1389–1430.  Google Scholar

  • [14]

    J. P. García Azorero and I. Peral Alonso, On limits of solutions of elliptic problems with nearly critical exponent, Comm. Partial Differential Equations 17 (1992), no. 11–12, 2113–2126.  CrossrefGoogle Scholar

  • [15]

    S. Kesavan, On two functionals connected to the Laplacian in a class of doubly connected domains, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003), no. 3, 617–624.  CrossrefGoogle Scholar

  • [16]

    K. Kurata and N. Shioji, Compact embedding from W01,2(Ω) to Lq(x)(Ω) and its application to nonlinear elliptic boundary value problem with variable critical exponent, J. Math. Anal. Appl. 339 (2008), no. 2, 1386–1394.  Web of ScienceGoogle Scholar

  • [17]

    P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoam. 1 (1985), no. 1, 145–201.  Google Scholar

  • [18]

    J. Liu, J.-F. Liao and C.-L. Tang, Ground state solutions for semilinear elliptic equations with zero mass in N, Electron. J. Differential Equations 2015 (2015), Paper No. 84.  Google Scholar

  • [19]

    S. Pohozaev, Eigenfunctions of the equation Δu+λf(u)=0, Soviet Math. Dokl. 6 (1965), 1408–1411.  Google Scholar

  • [20]

    G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 353–372.  CrossrefGoogle Scholar

  • [21]

    M. Willem, Minimax Theorems, Progr. Nonlinear Differential Equations Appl. 24, Birkhäuser, Boston, 1996.  Google Scholar

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.

Export Citation

© 2020 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

Comments (0)

Please log in or register to comment.
Log in