Generalized Concentration-Compactness Principles for Variable Exponent Lebesgue Spaces with Asymptotic Analysis of Low Energy Extremals

In this paper, we prove two generalized concentration-compactness principles for variable exponent Lebesgue spaces and as an application study the asymptotic behaviour of low energy extremals.


Introduction
The concentration-compactness principle (CCP) by Lions [1] has been a fundamental tool to study solutions of different kinds of elliptic PDEs with critical growth (in the sense of Sobolev embeddings), see [2][3][4][5] for some of its applications. Later on, in [6,7] Lions CCP was generalized by considering a general growth at infinity.
Let Ω be a bounded sub domain of R N , for an exponent p(x) we will use p − := inf x∈Ω p(x), p + := sup x∈Ω p(x) and p * (x) := N p(x) N−p(x) when p(x) < N. An exponent q(x) ≤ p * (x) is said to be critical if x ∈ C := {x ∈ Ω : q(x) = p * (x)}. In order to deal with the critical growth at infinity of the source function g that is |g(x, s)| ≤ c(1 + |s| q(x) ), (2) with q(x) ≤ p * (x), Bonder and Silva [13] and Yongqiang [14] extended Lions CCP to variable exponent settings, independently. Their method of proof followed the same lines as the ones that originated in Lions work. Let G : R → R be an upper semicontinuous, not zero in L 1 sense and satisfying the growth condition 0 ≤ G(s) ≤ c min |s| q + , |s| q − for s ∈ R, where p ≤ q ≤ p * , 1 < p − ≤ p(x) ≤ p + < N. This paper aims to study the Problem (1) with a general growth at infinity by extending the work of Flucher and Müller [6] to variable exponent Lebesgue spaces L p(x) (Ω) and W 1,p(x) (Ω). To be more precise, we study the concentration/compactness of the sequence G(v ) for v ∈ W 1,p(x) 0 (Ω) (closure of the set of test functions in variable exponent Sobolev space), whereas, Bonder and Silva [13] studied |v | q(x) . Thus, our work considerably contributes to the existing literature and it allows us to study Bernoulli's free-boundary problem, plasma problem and others in the variable exponent settings, see [7] for more details. We prove that in a extreme case either the sequence of measures concentrate to a dirac measure or have a convergent subsequence.
In addition, we analyse the asymptotic behaviour of solutions of the following variational problem, related to low energies when → 0. Problem (4) and its other variants for a constant exponent were rigorously studied, see [6,7,[15][16][17] and references therein. To establish the concentration or compactness of low energy extremals, another version of CCP is proved for the variable exponent Lebesgue spaces. When G is smooth i.e., G = g, solutions of (4) satisfy the following Dirichlet problem For a detailed study on nonlinear PDEs with variable exponent, we refer [18]. Organisation of this paper: Section 2 collects some necessary primary results to be used in later sections. Section 3 deals with the proof of generalized CCP and concentration/compactness result. Section 4 is committed to the variational problem of low energy extremals. Finally, Section 5 ends the manuscript with some concluding remarks.

Preliminary and Known Results
We present some preliminary concepts of variable exponent Lebesgue and Sobolev spaces. Let p : Ω → [1, ∞] be a measurable function and Ω be a bounded smooth subset of R N . Then L p(x) (Ω) is defined as In addition, p (x) = p(x)/(p(x) − 1) is known as conjugate exponent of p(x), further will be used throughout the paper and The exponent p(x) is called log-Hölder continuous if Let ρ(v) := Ω |v(x)| p(x) dx then the following proposition proved in [19] is quite useful.

Proposition 4 (Poincaré inequality). For all v in W
By the above proposition for W 1,p(x) 0 (Ω) both norms ∇v L p(x) (Ω) and v W 1,p(x) (Ω) are equivalent. Lastly, we present a localized sobolev type inequality from [14]. By B r (x) we mean a ball of radius r centered at x in Ω. Proposition 5. Take x 0 in Ω. For every δ > 0 there is a constant k(δ) independent of x in Ω such that if 0 < r < R with r R < k(δ) then there is a cut-off test function φ R r in W for all v in W 1,p(x) 0 (Ω).

Generalized Concentration-Compactness Principle
The exponent q(.) is critical when x ∈ C. The version of CCP proved in [14], only considered the critical case, whereas, in [13] q(.) was allowed to be subcritical as well. Later on, in [20] CCP was refined a bit to study immersion problem for the variable exponent Sobolev space. Now, we introduce some more notations in order to present the main results. •

By Sobolev embedding and Poincaré inequality for all
By Growth condition (3) and Inequality (15), we have (Ω). Now, we present the generalized CCP in form of following theorem. Let M(Ω) be a set of all nonnegative finite Borel measures on Ω and η * η in the sense of measure if Ω φη dx → Ω φηdx for all φ in C(Ω).

Theorem 1. Let p and q be log-Hölder continuous exponents with
(Ω), Then, for a countable index set J where {x j } j∈J ⊆ C, η is a positive nonatomic measure in M(Ω) and h ∈ L 1 (Ω). Moreover, atomic and regular parts satisfy the following generalized Sobolev type inequalities The strategy of the proof is analogous to that of [6,7], adapted to the variable exponent settings. In order to prove generalized CCP, first we prove two types of local generalized Sobolev inequalities, given in the following lemma. Lemma 1. Take δ > 0 and r < R satisfying r R ≤ k(δ) as in the Proposition 5. For x 0 ∈ Ω and G satisfying the growth condition (3) following inequalities hold Proof. Without loss, assume x 0 = 0 and let φ R r be a cutoff test function as in Proposition 5 i.e., Inequality (23) follows by letting R 1 → ∞, R 2 → ∞ in a way that R 2 /R 1 → ∞ and extending v as zero outside Ω. Now, we proceed to prove generalized CCP.
Proof of Theorem 1.
Step 3: (Decomposition of ζ) There is a subsequence such that |v | q(x) * ζ * in M(Ω). By the CCP ( [13], Theorem 1.1), we know that with x j ∈ C for all j ∈ J. By Growth condition (3) ζ ≤ cζ * and ζ is absolutely continuous with respect ζ * . Thus, by Radon-Nikodym theorem there is h in L 1 (Ω) such that Step 4: (Estimation of regular part) Fix δ > 0, choose j 0 such that ∑ j 0 <j η j < δ and take R in a way that B R (x j ) are disjoints for all j ≤ j 0 . Consider, where φ R r is a cutoff test function having support in B R (0), as in Proposition 5 and 0 < R 1 < R 2 . Then, support of ψ is in B R 2 (0)\ j≤j 0 B R (x j ). By Lemma 1 Taking R → 0, R 1 → ∞ and δ → 0, we get our desired generalized Sobolev type inequality for regular part.
Like in [13,14], generalized CCP can be used to prove the existence of solutions of different kinds of PDEs, but here we focus on the concentration/compactness of the maximizing sequence of generalized Sobolev constant G * i.e., Ω G(v )dx → G * when → 0.
As we know ζ(Ω) ≤ G * , but when we have equality, then there are two possibilities, either the limit measure is non-atomic, or the sequence concentrates to a single point, see the following result. Our idea is to use a type of convexity argument to prove it.

Generalized Concentration Compactness Principle for Low Energies
In a model without external energy source, internal energies will run out eventually. We deal with possible limit of low energy extremals of (4) and determine its shape. For v in W 1,p(x) 0 (Ω) with ∇v L p(x) (Ω) ≤ , consider w := v/ then by Growth condition (3) and Sobolev embedding (15) Theorem 3. Let p and q be log-Hölder continuous exponents with (Ω), ζ in the sense of measure in M(Ω).
Then, for a countable index set J where {x j } j∈J ⊆ C, η is a positive nonatomic measure in M(Ω) and h ∈ L 1 (Ω). Moreover, atomic and regular parts satisfy the following generalized Sobolev type inequalities In order to prove generalized CCP of low energies, first we prove couple of auxiliary lemmas.

Lemma 2.
Consider G * (p(.), q(.), R N ) and < 1. Then: , then w is admissible for G * and we have taking supremum for all such v gives us (a) and On the other hand fix δ > 0, x 0 in Ω. There exists w in W 1,p(x) 0 (Ω) such that For sufficiently large r > 0 By Proposition 5 there exist a cutoff test function φ R r , supported in B R (0) with for sufficiently small , v is supported in Ω and ∇v L p(x) (Ω) ≤ . Now, (Ω) with ∇v L p(x) (Ω) ≤ . Take w = v/ , δ > 0 and r < R satisfying r R ≤ k(δ) as in the Proposition 5. For x 0 ∈ Ω and G satisfying the growth condition (3), following inequalities hold Proof. The proof is similar as of Lemma 1.
The generalized CCP for low energies is proved in the same manner of Theorem 1.

Proof of Theorem 3.
Steps 1-4 are analogous with the use of Lemmas 2 and 3, as in proof of Theorem 1, we just need to prove pointwise estimate (32) and Inequality (33) for the regular part. Indeed, there exists a subsequence such that For A ⊆ Ω and s > 0 However, as we know lim s→∞ |{x ∈ A : |w | ≥ s}| = 0, it yields that By Radon-Nikodym theorem, we deduce that h ≤ G + 0 |w| q(x) a.e. in Ω. Lastly, Inequality (33) follows from integration and Sobolev inequality (15).
Lastly, we know ζ(Ω) ≤ G * , but in case of equality and G − 0 = G + 0 , in comparison to Theorem 2 compactness of low energies results into an approximation of Sobolev constant S i.e., If in addition G + 0 = G − 0 then w → w in W 1,p(x) 0 (Ω), G * = G 0 S and w is an extremal for S.
By Growth condition (3) Therefore, by Dunford-Pettis compactness theorem for a subsequence in comparison with Theorem 2, convergence is not strong, the main reason for which is L 1 norm is not strictly convex. Let that G + 0 = G − 0 = G 0 > 0, G 0 cannot be zero. By Fatou's lemma, we have G − 0 S ≤ G * and together with Inequality (33) yield The equality in above implies ∇w L p(x) (Ω) = 1, therefore η = |w| p(x) and w → w strongly in Therefore, taking s → ∞

Conclusions and Future Work
The main work of this paper is to study a class of elliptic equations with general growth at infinity for variable exponent Lebesgue spaces. The main results nicely determine the limit measures. Therefore, with the proposed work, we can study several models in variable exponent settings, for many fields like plasma physics, fluid mechanics and control systems. As future lines of research, one can also explored concentration compactness principles for fractional Sobolev spaces and fractional PDEs, we refer [21][22][23][24][25][26] for basic theory. One can also study the convergence of low energy extremals with variational methods.