## Abstract

In this paper, we first establish a principle of concentration compactness in . Then, based on this concentration compactness principle, we study the existence of solutions for a class of *p*(*x*)-Laplacian equations in involving the critical exponent. Under suitable assumptions, we obtain a sequence of radially symmetric solutions associated with a sequence of positive energies going towards infinity.

## 1. Introduction and main result

With the emergence of nonlinear problems in natural science and engineering, Sobolev spaces *W*^{1, p} demonstrate their limitations in applications. A class of nonlinear problems with variable exponential growth, for example, is a new research field and reflects a new kind of physical phenomena.

Since the spaces *L*^{p(x)} and *W*^{1, p(x)} were thoroughly studied by Kováčik & Rákosník (1991), variable exponent Sobolev spaces have been used in the last decades to model various phenomena. In the studies of a class of nonlinear problems, variable exponent Sobolev spaces play an important role. Ruzicka (2000) presented the mathematical theory for the application of variable exponent Sobolev spaces in electro-rheological fluids. In recent years, differential equations and variational problems with *p*(*x*)-growth conditions have been studied extensively (e.g. Alves & Souto 2005; Chabrowski & Fu 2005; Mihăilescu & Rădulescu 2006; Antontsev *et al.* 2007).

Equations involving critical Sobolev exponents have received great attention ever since the seminal work of Brezis & Nirenberg (1983). Lions (1985) established the concentration compactness principle of the limit case in the calculus of variations. Then it was widely used in the study of partial differential equations with critical growth (e.g. Egnell 1988; Drabek & Huang 1997; Goncalves & Alves 1998; Li & Zhang 2009, and the references therein). Fu (2009) established a principle of concentration compactness in Sobolev space *W*^{1, p(x)}(*Ω*), where is a bounded domain, and then discussed the existence of solutions for a class of *p*(*x*)-Laplacian equations with critical growth.

In this paper, we will establish a principle of concentration compactness in , then apply it to obtain the existence of solutions for the following kind of *p*(*x*)-Laplacian equations with critical exponent:
1.1
where *p* is Lipschitz continuous, radially symmetric on and satisfies 1<*p*_{−}≤*p*(*x*)≤*p*_{+}<*N*, *p**(*x*)=*N**p*(*x*)/(*N*−*p*(*x*)), *λ*>0. Here we denote
and denote by *p*_{1}(*x*)≪*p*_{2}(*x*) the fact that

First, we recall some basic properties of variable exponent Lebesgue spaces *L*^{p(x)}(*Ω*) and variable exponent Sobolev spaces *W*^{1, p(x)}(*Ω*), where is a domain. For a deeper treatment of these spaces, we refer to Edmunds *et al.* (1999), Edmunds & Rákosník (2000), Fan *et al.* (2001*a*,*b*) and Kováčik & Rákosník (1991).

Let ** P**(

*Ω*) be the set of all Lebesgue measurable functions and 1.2 The variable exponent Lebesgue space

*L*

^{p(x)}(

*Ω*) is the class of all functions

*u*such that Under the assumption

*L*

^{p(x)}(

*Ω*) is a Banach space equipped with the norm (1.2).

The variable exponent Sobolev space *W*^{1, p(x)}(*Ω*) is the class of all functions *u*∈*L*^{p(x)}(*Ω*) such that |∇*u*|∈*L*^{p(x)}(*Ω*). For *u*∈*W*^{1, p(x)}(*Ω*), we define
1.3
then ||*u*|| is a norm on *W*^{1, p(x)}(*Ω*).

Throughout this paper, we assume that the following conditions hold:

(H1) and satisfies |

*f*(*x*,*t*)|≤*g*(*x*)|*t*|^{α(x)−1}, where , 1<*α*_{−}≤*α*(*x*)≪*p*(*x*) or*p*(*x*)≪*α*(*x*)≪*p**(*x*) and ,*q*(*x*)=*p**(*x*)/(*p**(*x*)−*α*(*x*)).(H2)

*f*(*x*,*t*)=−*f*(*x*,−*t*) for any .(H3)

*f*(*x*,*t*)=*f*(|*x*|,*t*) for any .(H4) , where

*h*is radially symmetric, i.e.*h*(*x*)=*h*(|*x*|) for any , and*h*(*x*)≥0(≢0) and .

Moreover, we assume that *f*(*x*,*t*) satisfies the following condition:

(H5) If

*α*(*x*) in (H1) satisfies*p*(*x*)≪*α*(*x*)≪*p**(*x*) , then there exists*μ*(*x*)≫*p*(*x*) such that 0≤*μ*(*x*)*F*(*x*,*t*)≤*f*(*x*,*t*)*t*, for any where

In this paper, our main result is the following.

## Theorem 1.1.

*Assume hypotheses H1–H5 are fulfilled. Then there exists λ*_{*}*>0 such that for any* *, problem (1.1) has a sequence of radially symmetric solutions* *such that*
*as*

## 2. Principle of concentration compactness

In this section, we will establish the principle of concentration compactness in .

## Definition 2.1.

Let *Ω* be an open subset of and define
and
The space *C*_{0}(*Ω*) is the closure of *K*(*Ω*) in *B**C*(*Ω*) with respect to the uniform norm . A finite measure on *Ω* is a continuous linear functional on *C*_{0}(*Ω*). The norm of the finite measure *μ* is defined by
where . We denote by *M*(*Ω*) the space of finite non-negative Borel measures on *Ω*. A sequence weakly-* in *M*(*Ω*) is defined by , for any *η*∈*C*_{0}(*Ω*).

## Theorem 2.2.

*Let* *with ||u*_{n}*||≤1 such that*
and
*as* *. Denote* *. Then, the limit measures are of the form*
and
*where J is a countable set,* *and* *is a non-atomic non-negative measure. The atoms and the regular part satisfy the generalized Sobolev inequality*
*and*
*where* *and*

In order to obtain theorem 2.2, we first give some lemmas.

## Lemma 2.3.

*Let For any δ>0, there exists k(δ)>0 independent of x such that for 0<r<R with r/R≤k(δ), there is a cut-off function with in B_{r}(x), outside B_{R}(x), and for any* ,

## Lemma 2.4.

*Let , δ>0 and r/R<k(δ), where k(δ) is from lemma 2.3. Then, for any we have*

Similar to the discussion of lemma 3.1 and corollary 3.1 of Fu (2009), it is easy to obtain the proof of lemma 2.3 and 2.4.

## Proof of theorem 2.2.

(i) and In fact, for any *R*>0, let such that 0≤*η*≤1; *η*≡1 in *B*_{R}(0). We obtain
Note that if ||*u*_{n}||≤1, then
Thus, we have . Let , then we obtain . As , similarly we could obtain .

(ii) , where *J* is a countable set, , and is a non-atomic non-negative measure.

Let . Denote
We know that *F* is convex and continuously differentiable on , so it is also weakly sequentially lower semicontinuous. We obtain
From it is immediate that
Then i.e. *μ*≥|∇*u*|^{p(x)}+|*u*|^{p(x)}. Thus, we have

(iii) , where is as above and As weakly in , passing to a subsequence, still denoted by {*u*_{n}}, we may assume that a.e. in .

Let . Note that and {*u*_{n}} is bounded in , then similar to lemma 2.10 of Fu (2009), we can easily obtain
Denote . As , we obtain
Thus, weakly-* in , it is easy to obtain that

In the following, we will verify that every atom of *ν* is that of *μ* and

Let . By lemma 2.4, for any *δ*>0, there exists *k*(*δ*)>0 such that for 0<*r*<*R* with *r*/*R*≤*k*(*δ*),
For any 0<*r*′<*r*, *R*′>*R*. Let such that 0≤*η*_{1}≤1; *η*_{1}≡1 in *B*_{r′}(*x*_{0}), such that 0≤*η*_{2}≤1; *η*_{2}≡1 in *B*_{R}(*x*_{0}). We obtain
Let , then we obtain
Thus,
where is the closure of *B*_{R′}(*x*_{0}). If , then . Thus, we have
Then, the atom of *ν* is that of *μ*.

Denote . Note that weakly in . Passing to a subsequence, still denoted by we may assume that there exists such that weakly-* in .

For any 0<*r*<*R*, let such that 0≤*η*≤1; *η*≡1 in *B*_{r}(*x*_{0}). Then by the definition of *C**, we obtain
Note that
We know in *L*^{p(x)}(*B*_{R}(*x*_{0})); then , as . We obtain
then
Let , then we have In particular, is absolutely continuous with respect to . By the Radon–Nikodym Theorem, there exists such that .

Note that *p* is Lipschitz continuous on . Thus, for any , there exists *R*_{0}>0 such that when *R*<*R*_{0}, we have
Similar to the above discussion, we could obtain
where and . By the Lebesgue differentiation theorem,

For any . If *k*(*x*)≠0, we obtain , then . Note that if and *ν* have the same atom, then *x* is an atom of *μ*, which is a contradiction. Thus, *k*≡0 in . Therefore, in . Note that is non-atomic, thus .

(iv) Let such that 0≤*η*≤1; *η*≡1 in *B*_{R}(0). Similar to the discussion in (iii), we obtain
Let , then we have
Note that , and let , we obtain the conclusion. ■

Theorem 2.2 does not provide any information about the possible loss of mass at infinity of a weakly convergent sequence. The following theorem 2.5 expresses this fact in quantitative terms.

## Theorem 2.5.

*Let* *such that* *weakly-* in* *,* *weakly-* in* *and define*
*and*
*The quantities* *and* *are well defined and satisfy*
and

## Proof.

Let such that 0≤*χ*≤1; *χ*≡1 in , *χ*≡0 in *B*_{1}(0). For any *R*>0, define *χ*_{R}(*x*)=*χ*(*x*/*R*). We have
then Similarly, we obtain
Note that
It is easy to verify that , as . Thus, we have

We obtain Similarly, we obtain ■

## 3. Proof of theorem 1.1.

In this section, we will discuss equation (1.1) when *p*(*x*) is radially symmetric.

First of all, let us introduce some notation. Let *O*(*N*) be the group of orthogonal linear transformations in , and *G* is a subgroup of *O*(*N*). For *x*≠0, we denote the cardinality of by |*G*_{x}| and set . An open subset *Ω* of is *G*-invariant if for any .

## Definition 3.1.

Let *Ω* be a *G*-invariant open subset of . The action of *G* on *W*^{1, p(x)}(*Ω*) is defined by for any *u*∈*W*^{1, p(x)}(*Ω*). The subspace of invariant functions is defined by
A functional is *G*-invariant if for any .

## Definition 3.2.

We say that is a weak solution of problem (1.1), that is, for any

To discuss problem (1.1), we need to define two functionals on :
and
where *λ*>0. In this section, for we define
3.1
then |||*u*||| is a norm that is equivalent with the norm defined by equation (1.3).

It is easy to check that and the weak solution for problem (1.1) coincides with the critical point of the function *φ*. Throughout this section, we denote by *c*_{i} the positive constants.

In the following, we assume *G*=*O*(*N*) and denote . By conditions (H3) and (H4), it is immediate that *φ* is *O*(*N*)-invariant. Then, by the principle of symmetric criticality of Krawcewicz & Marzantowicz (1990), we know that *u*_{0} is a critical point of *φ* if and only if *u*_{0} is a critical point of . Therefore, it suffices to prove the existence of a sequence of critical points for on .

## Lemma 3.3.

*There exists λ_{*}>0 such that when any PS sequence i.e. and as is bounded*.

## Proof.

Note that if *p* is radially symmetric on , then there exists a Lipschitz continuous function *ν*(*x*) such that *p*(*x*)≪*ν*(*x*)≤*p**(*x*) and radially symmetric on , which will be given in the following. Denote we obtain
By the Young inequality, we can obtain that for any *ε*∈(0,1), there exists *C*(*ε*)>0 such that
Thus,

(i) If *p*(*x*)≪*α*(*x*)≪*p**(*x*). Take
Let *ε*≤*l*_{0}/2 and *λ*_{*}=2*C*(*ε*)/*l*_{0}. When *λ*≥*λ*_{*}, by condition (H5), we have
Then {*u*_{n}} is bounded.

(ii) If 1<*α*_{−}≤*α*(*x*)≪*p*(*x*). Take *ν*(*x*)=*p**(*x*). By condition (H1), we obtain that for any , |*F*(*x*,*t*)|≤(*g*(*x*)/*α*(*x*))|*t*|^{α(x)}. Similar to lemma 4.3 of Fu & Zhang (2009), we obtain that there exists *c*_{0}>0 such that
Let *ε*≤*l*_{0}/4 and *λ*_{*}=4*C*(*ε*)/*l*_{0}. When *λ*≥*λ*_{*}, we have
Then {*u*_{n}} is bounded. ■

## Lemma 3.4.

*When λ≥λ_{*}, any PS sequence contains a convergent subsequence, where λ_{*} is from lemma 3.3*.

## Proof.

Let be a PS sequence. By lemma 3.3, we obtain that {*u*_{n}} is bounded when *λ*≥*λ*_{*}. As is reflexive, passing to a subsequence, still denoted by {*u*_{n}}, we may assume that there exists such that weakly in and a.e. on . We can also obtain that weakly on , as .

Note that It is easy to obtain By condition (H1), similar to theorem 4.3 of Fu & Zhang (2009), we can also obtain as . If we could verify that we can easily obtain as . Thus, we have as Then similar to theorem 3.1 of Fu (2007), we obtain i.e. in

In the following, we will verify that
Note that weakly in . Passing to a subsequence, still denoted by {*u*_{n}}, we may assume that there exist such that and weakly-* in . By theorem 2.2, we obtain and where *J* is a countable set, , and is a non-atomic positive measure. Then by theorem 2.5, we obtain

(i) For any *j*∈*J*, *ν*_{j}=0. Suppose that there exists *x*_{j 0}≠0, where *j* _{0}∈*J* such that *ν*_{j 0}=*ν*({*x*_{j 0}})>0. As , the measure *ν* is *O*(*N*)-invariant. For any , . We know that , thus . Note that the measure *ν* is finite, which is a contradiction. Then, for any *x*_{j}≠0, where *j*∈*J*, we obtain *ν*_{j}=*ν*({*x*_{j}})=0.

0∉{*x*_{j}:*j*∈*J*}. For any *ε*>0, take a radially symmetric function such that 0≤*ϕ*≤1, |∇*ϕ*|≤2/*ε*; *ϕ*≡1 on *B*_{ε}(0). It is easy to obtain {*u*_{n}*ϕ*} is bounded on , then we have , as . Note that
By condition (H1), we have that |*f*(*x*,*t*)|≤*g*(*x*)|*t*|^{α(x)−1}, where , 1<*α*_{−}≤*α*(*x*)≪*p*(*x*) or *p*(*x*)≪*α*(*x*)≪*p**(*x*). We know in *L*^{α(x)}(*B*_{2ε}(0)), thus we obtain as . Then, we obtain
It is easy to verify that in *L*^{p(x)}(*B*_{2ε}(0)), thus , as . Then, we obtain
Note that
and
where *ω*_{N} is the surface area of the unit sphere in . As , we obtain , as . Similarly, we can also obtain
as . Thus, 0=−*μ*({0})+*h*(0)*ν*({0}). Note that if *h*(0)=0, then *μ*({0})=0, i.e. 0 is not an atom of *μ*.

(ii) For any *R*>0, take a radially symmetric function such that 0≤*χ*_{R}≤1, |∇*χ*_{R}|≤2/*R*; *χ*_{R}≡1 in , *χ*_{R}≡0 in *B*_{R}(0). It is easy to obtain that {*u*_{n}*χ*_{R}} is bounded on , then , as . We have
Note that weakly in , then As
by condition (H1), similar to theorem 4.3 of Fu & Zhang (2009), for any *ε*>0, there exists *R*_{1}>0 such that when *R*>*R*_{1}, for any . Note that
as . Thus, we obtain that
Note that , then for any *ε*>0, there exists *R*_{2}>0 such that when *R*>*R*_{2}, we have Then,
Similar to the discussion in (i), we obtain that , as . It is easy to obtain that in , thus as . Note that if
then We have
thus . We know that
and ; it is easy to obtain that By the imbedding theorem, for any , |*u*|_{p*(x)}≤*c*_{8}|||*u*|||. Then for any , there exists *R*_{3}>0 such that when *R*≥*R*_{3},
We obtain that there exists such that when *k*≥*k*_{0},
Then, we obtain that
and |*χ*_{R}*u*_{n}|_{p*(x)}<*c*_{8}⋅*ε*^{1/p+}<1, for any *n*≥*k*. It is easy to obtain that
thus

By (i) and (ii), we obtain that
Passing to a subsequence, still denoted by {*u*_{n}}, we have
as . Note that if , then by the Fatou lemma, we have
Thus,
i.e. in . Note that , it is easy to obtain that , as ■

As is a separable and reflexive Banach space, there exist and such that and and where is the dual of .

In the following, we will denote for *k*=1,2,…. Then we obtain the following lemma.

## Lemma 3.5.

*There exist τ_{k}>0 and ρ_{k}>0, such that for any if |||u|||=ρ_{k}, when is sufficiently large*.

## Proof.

For and |||*u*|||≥1, we have
Denote
and
By lemma 3.3 of Fan & Han (2004), we obtain that as

In the following, we will verify that if as It is obvious that 0≤*γ*_{k+1}≤*γ*_{k}, then , as . There exist with |||*u*_{k}|||=1 such that
for each *k*=1,2,…. As is reflexive, passing to a subsequence, still denoted by {*u*_{k}}, we may assume that there exists such that weakly in , as .

We claim that *u*=0. In fact, for any *f*_{m}∈{*f*_{n}:*n*=1,…,*m*,…}, we have *f*_{m}(*u*_{k})=0 when *k*>*m*; then , as . It is immediate that for any , *f*_{m}(*u*)=0. Note that if then we have *u*=0.

By theorem 2.2, there exist finite measure *ν* and sequences such that weakly-* in , where *J* is a countable set. Similar to the discussion in lemma 3.4(i), we obtain *ν*_{j}=*ν*({*x*_{j}})=0 for any *j*∈*J* and *x*_{i}≠0.

For any 0<*r*<*R*, take such that 0≤*η*≤1; *η*≡1 in *B*_{R}(0)\*B*_{r}(0), *η*≡0 in *B*_{r/2}(0). Then,
as . Note that if , then

As , for any *ε*>0, there exist *r*_{0}, *R*_{4}>0 such that when |*x*|≤*r*_{0}, *h*(*x*)<*ε*; when |*x*|≥*R*_{4}, *h*(*x*)<*ε*. Thus, for any , we have
and
where *r*<*r*_{0}, *R*>*R*_{4}. We obtain Then , as . Thus, we obtain
Let Note that as *k* is sufficiently large, Then, for any |||*u*|||=*ρ*_{k},
It is easy to obtain that as ■

By condition (H4), there exists such that *h*(*x*_{0})>0. Then there exist 0<*r*_{1}<*r*_{2} such that *r*_{1}<|*x*_{0}|<*r*_{2},
and for any |*x*|∈(*r*_{1},*r*_{2}) and *h*(*x*)≥*h*(*x*_{0})/2. Take radially symmetric functions *i*=1,…,*k*, satisfying for *i*≠*j*. From
it is immediate that codim dim for any

## Lemma 3.6.

*For any there exists R_{k}>0 such that for any*

*if*|||

*u*|||≥

*R*

_{k}, .

## Proof.

For and |||*u*|||≥1, we have that
By the Young inequality, for any *ε*>0, there exists *c*(*ε*)>0 such that
Take , thus
We know that |⋅|_{p*(x)} is also a norm on and is a finite-dimensional space; then |||⋅||| and |⋅|_{p*(x)} are equivalent. Thus, there exists *c*_{10}>1 such that for any , we have |||*u*|||≤*c*_{10}|*u*|_{p*(x)}. When |||*u*|||>*c*_{10},
As we can obtain that there exists *R*_{k}>0, such that for any : if |||*u*|||≥*R*_{k}, ■

## Proof of theorem 1.1.

By condition (H2), is an even function on . From lemmas 3.3–3.6 and by theorem 6.3 of Struwe (1996), we know that when *λ*≥*λ*_{*} and is sufficiently large,
is a critical value for , and *w*_{k}≥*τ*_{k}, where , : *f* is odd, *f*(*u*)=*u*, if and |||*u*|||≥*R*_{k}} and *λ*_{*}>0 is from lemma 3.3. By lemma 3.5, if then as We obtain that function has a sequence of critical points such that as ■

## Footnotes

- Received September 3, 2009.
- Accepted November 19, 2009.

- © 2010 The Royal Society