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Global compactness for a class of quasi-linear elliptic problems

We prove a global compactness result for Palais-Smale sequences associated with a class of quasi-linear elliptic equations on exterior domains.


Introduction and main result
Let Ω be a smooth domain of R N with a bounded complement and N > p > m > 1. The main goal of this paper is to obtain a global compactness result for the Palais-Smale sequences of the energy functional associated with the following quasi-linear elliptic equation (1.1) − div(L ξ (Du)) − div(M ξ (u, Du)) + M s (u, Du) + V (x)|u| p−2 u = g (u) in Ω, where u ∈ W 1,p 0 (Ω) ∩ D 1,m 0 (Ω), meant as the completion of the space D(Ω) of smooth functions with compact support, with respect to the norm u W 1,p (Ω)∩D 1,m (Ω) = u p + u m , having set u p := u W 1,p (Ω) and u m := Du L m (Ω) . We assume that V is a continuous function on Ω, As known, lack of compactness may occur due to the lack of compact embeddings for Sobolev spaces on Ω and since the limiting equation on R N where ∆ p u := div(|Du| p−2 Du), for a suitable function a ∈ C 1 (R; R + ), or the even simpler case where a is constant, namely (1.4) − ∆ p u − ∆ m u + V (x)|u| p−2 u = |u| σ−2 u in Ω.
Since the pioneering work of Benci and Cerami [2] dealing with the case L(ξ) = |ξ| 2 /2 and M (s, ξ) ≡ 0, many papers have been written on this subject, see for instance the bibliography of [12]. Quite recently, in [12], the case L(ξ) = |ξ| p /p and M (s, ξ) ≡ 0 was investigated. The main point in the present contribution is the fact that we allow, under suitable assumptions, a quasi-linear term M (u, Du) depending on the unknown u itself. The typical tools exploited in [2,12], in addition to the point-wise convergence of the gradients, are some decomposition (splitting) results both for the energy functional and for the equation, along a given bounded Palais-Smale sequence (u n ). To this regard, the explicit dependence on u in the term M (u, Du) requires a rather careful analysis. In particular, we can handle it for ν|ξ| m ≤ M (s, ξ) ≤ C|ξ| m , p − 1 ≤ m < p − 1 + p/N.
The restriction on m, together with the sign condition (1.9) provides, thanks to the presence of L, the needed a priori regularity on the weak limit of (u n ), see Theorems 3.2 and 3.4. Besides the aforementioned motivations, which are of mathematical interest, it is worth pointing out that in recent years, some works have been devoted to quasi-linear operators with double homogeneity, which arise from several problems of Mathematical Physics. For instance, the reaction diffusion problem u t = −div(D(u)Du) + ℓ(x, u), where D(u) = d p |Du| p−2 + d m |Du| m−2 , d p > 0 and d m > 0, admitting a rather wide range of applications in biophysics [10], plasma physics [16] and in the study of chemical reactions [1]. In this framework, u typically describes a concentration and div(D(u)Du) corresponds to the diffusion with a coefficient D(u), whereas ℓ(x, u) plays the rǒle of reaction and relates to source and loss processes. We refer the interested reader to [5] and to the reference therein. Furthermore, a model for elementary particles proposed by Derrick [9] yields to the study of standing wave solutions ψ(x, t) = u(x)e iωt of the following nonlinear Schrödinger equation for which we refer the reader e.g. to [3].
In order to state the first main result, assume N > p > m ≥ 2 and and consider the C 2 functions L : R N → R and M : R × R N → R such that both the functions ξ → L(ξ) and ξ → M (s, ξ) are strictly convex and for all ξ ∈ R N . Furthermore, we assume for all (s, ξ) ∈ R × R N and that the sign condition (cf. [14]) holds for all (s, ξ) ∈ R × R N . Also, G : R → R is a C 2 function with G ′ (s) := g(s) and for all s ∈ R. We define See Section 2 for some properties of the functionals φ and φ ∞ .
The first main global compactness type result is the following Theorem 1.1. Assume that (1.5)-(1.11) hold and let (u n ) ⊂ W 1,p 0 (Ω) ∩ D 1,m 0 (Ω) be a bounded sequence such that as well as Let us now come to a statement for the cases 1 < m ≤ 2 or 1 < p ≤ 2. Let us define If either p < 2, σ < 2 or m < 2, we shall weaken the twice differentiability assumptions, by and M ss ∈ C 0 (R × R N ). Moreover we assume the same growth conditions for L, M, G and their derivatives, replacing only the growth assumptions for L ξξ , M ξξ , G ′′ by the following hypotheses: (1.14) Conditions (1.12)-(1.13), in some more concrete situations, follow immediately by homogeneity of L ξ and G ′ (see, for instance, [12, Lemma 3.1]). Similarly, (1.14) is satisfied for instance when M is of the form M (s, ξ) = a(s)µ(ξ), being a : R → R + a bounded function and µ : R N → R + a C 1 strictly convex function such that µ ξ is homogeneous of degree m − 1.
Theorem 1.2. Under the additional assumptions (1.12)-(1.14) in the sub-quadratic cases, the assertion of Theorem 1.1 holds true.
As a consequence of the above results we have the following compactness criterion.
and S p,σ and C g are constants such that S p,σ u σ p ≥ u σ L σ (R N ) and |g(s)| ≤ C g |s| σ−1 . Remark 1.4. It would be interesting to get a global compactness result in the case L = 0 and p = m, namely for the model case Notice that, even assuming a ′ bounded, a ′ (u)|Du| m is merely in L 1 (Ω) for W 1,m 0 (Ω) distributional solutions. In general, in this setting, the splitting properties of the equation are hard to formulate in a reasonable fashion. Remark 1.6. We prove the above theorems under the a-priori boundedness assumption of (u n ). This occurs in a quite large class of problems, as Proposition 2.2 shows. Remark 1.7. With no additional effort, we could cover the case where an additional term W (x)|u| m−2 u appears in (1.1) and the functional framework turns into W 1,p 0 (Ω) ∩ W 1,m 0 (Ω). In the spirit of [11], we also get the following with V (x) → 1 as |x| → ∞. Assume furthermore that Then (1.16) admits a minimizer.
Remark 1.9. We point out that, some conditions guaranteeing the nonexistence of nontrivial solutions in the star-shaped case Ω = R N can be provided. For the sake of simplicity, assume that L is p-homogeneous and that ξ → M (s, ξ) is m-homogeneous. Then, in view of [13,Theorem 3], that holds for C 1 solutions by virtue of the results of [8], we have that (1.1) admits no nontrivial C 1 solution well behaved at infinity, namely satisfying condition (19) of [13], provided that there exists a number a ∈ R + such that a.e. in R N and for all (s, ξ) ∈ R × R N (N − p(a + 1))L(ξ) + (N − m(a + 1))M (s, ξ) + (asg(s) − N G(s)) holding, for instance, if there exists 0 ≤ a ≤ N −p p such that asg(s) − N G(s) ≥ 0, for a.e. x ∈ R N and for all (s, ξ) ∈ R × R N . Also, in the more particular case where g(s) = |s| σ−2 s and V (x) = V ∞ > 0, then the above conditions simply rephrase into for every (s, ξ) ∈ R × R N . In fact, in (1.9), we consider the opposite assumption on M s .

Some preliminary facts
It is a standard fact that, under condition (1.6) and (1.10), the functionals Analogously, although M depends explicitly on s, the functional admits, thanks to condition (1.5), directional derivatives along any v ∈ W 1,p 0 (Ω) ∩ D 1,m 0 (Ω) and as it can be easily verified observing that p ≤ p p−m ≤ p * and that, for u ∈ W 1,p 0 (Ω) ∩ D 1,m 0 (Ω), by Young's inequality, for some constant C it holds This yields the desired convergence, using (1.7) and the Dominated Convergence Theorem. Notice that the same argument carried out before applies either to integrals defined on Ω or on R N . Hence the following proposition is proved.
In the hypotheses of Theorems 1.1 and 1.2, the functionals φ and φ ∞ are C 1 .
In addition to the assumptions on L, M and g, G set in the introduction, assume now that there exist positive numbers δ > 0 and µ > p such that for any s ∈ R and all ξ ∈ R N . This hypothesis is rather well established in the framework of quasi-linear problems (cf. [14]) and it allows an arbitrary Palais-Smale sequence (u n ) to be bounded in W 1,p 0 (Ω) ∩ D 1,m 0 (Ω), as shown in the following Proposition 2.2. Let j be as in (1.11) and assume that ( recalling the definition of j, and using condition (2.1), yields The assertion then follows immediately.
From now on we shall always assume to handle bounded Palais-Smale sequences, keeping in mind that condition (2.1) can guarantee the boundedness of such sequences. Proposition 2.3. Let j be as in (1.11) and assume that 1 < m < p < N and p < σ < p * . Let Then, up to a subsequence, (u n ) converges weakly to some u in Proof. It is sufficient to justify that Du n (x) → Du(x) for a.e. x ∈ Ω. Given an arbitrary bounded Due to the strict convexity assumptions on the maps ξ → L(ξ) and ξ → M (s, ξ) and the growth conditions on L ξ , M ξ , M s and g, all the assumptions of [6, Theorem 1] are fulfilled. Precisely, for a.e. x ∈ ω and all (s, ξ) ∈ R × R N , and Then, it follows that Du n (x) → Du(x) for a.e. x ∈ ω. Finally, a simple Cantor diagonal argument allows to recover the convergence over the whole domain Ω.
Next we prove a regularity result for the solutions of equation (1.1).
Then h(x, u n , Du n ) converges to h(x, u, Du) in L µ (Ω).
We have the following splitting result Theorem 3.2. Let the integrand j be as in (1.11) and Assume that (u n ) ⊂ W 1,p 0 (Ω) ∩ D 1,m 0 (Ω) is a bounded Palais-Smale sequence for φ at the level c ∈ R weakly convergent to some u ∈ W 1,p 0 (Ω) ∩ D 1,m 0 (Ω). Then being u n and u regarded as elements of W 1,p (R N ) ∩ D 1,m (R N ) after extension to zero out of Ω.

Remark 3.3.
In order to shed some light on the restriction (1.5) of m, it is readily seen that it is a sufficient condition for the following local compactness property to hold. Assume that ω is a smooth domain of R n with finite measure. Then, if (u h ) is a bounded sequence in W 1,p 0 (ω), there exists a subsequence (u h k ) such that where Υ(x, s, ξ) = g(s)−M s (s, ξ)−V (x)|s| p−2 s. In fact, taking into account the growth condition on g and M s , this can be proved observing that, for every ε > 0, there exists C ε such that we refer the reader to [12].
Proof. Fixed some v ∈ W 1,p 0 (Ω) ∩ D 1,m 0 (Ω), let us define for a.e. x ∈ Ω and all (s, ξ) ∈ R × R N , In order to prove 3.5 we are going to show that (3.6) lim As it can be easily checked, there holds Hence, by plugging the particular form of j in the above equation yields where a(x, s, ξ) : We claim that, as n → ∞, it holds a(·, u n , Du n ) → a(·, u, Du) Then, using Hölder's inequality and the embeddings of W 1,p 0 (Ω)∩D 1,m 0 (Ω) into L σ (Ω) and L p * (Ω) we obtain yielding the desired conclusion (3.6). It remains to prove the convergences we claimed above. For each term, we shall exploit Lemma 2.5. Since m < p − 1 + p/N , we can set it follows β > 0 and m < m + α < p. Young's inequality yields in turn y (m+α)/(p * ) ′ ≤ Cy m/(p * ) ′ + Cy p/(p * ) ′ , for all y ≥ 0.
Theorem 3.5. Assume (1.9), let the integrand j be as in (1.11) and p ≤ 2 or m ≤ 2 or σ ≤ 2, Assume that (u n ) ⊂ W 1,p 0 (Ω) ∩ D 1,m 0 (Ω) is such that u n ⇀ u, u n → u a.e. in Ω, Du n → Du a.e. in Ω and there exists (w n ) in (W 1,p 0 (Ω) ∩ D 1,m 0 (Ω)) * such that w n → 0 as n → ∞ and, for every v ∈ W 1,p 0 (Ω) ∩ D 1,m 0 (Ω), Then φ ′ (u) = 0. Moreover, there exists a sequence (ξ n ) that goes to zero in (W 1,p 0 (Ω) ∩ D 1,m 0 (Ω)) * , such that Keeping in mind the argument in proof of Theorem 3.4, here we shall be more sketchy. For every s ∈ R and ξ ∈ R N we plug L, M, G into the equation We write the term M ξ (s − u(x), ξ − Du(x)) − M ξ (s, ξ) in a more suitable form, namely , so that The term a ′ admits the same growth condition of a, cf. the proof of Theorem 3.4. Also, since as for the term c 1 in the proof of Theorem 3.4 we obtain On the other hand, directly from assumptions (1.12)-(1.14) we get The conclusion follows then by the same argument carried out in Theorem 3.4.
We can finally come to the proof of the main results.

Proof of Theorems 1.1 and 1.2 completed
We follow the scheme of the proof given in [17, p.121]. Let (u n ) ⊂ W 1,p 0 (Ω) ∩ D 1,m 0 (Ω) be a bounded Palais-Smale sequence for φ at the level c ∈ R. Hence, there exists a sequence (w n ) in the dual of W 1,p 0 (Ω) ∩ D 1,m 0 (Ω) such that w n → 0 and φ(u n ) → c as n → ∞ and, for all v ∈ W 1,p 0 (Ω) ∩ D 1,m 0 (Ω), we have Since (u n ) is bounded in W 1,p 0 (Ω) ∩ D 1,m 0 (Ω), up to a subsequence, it converges weakly to some function v 0 ∈ W 1,p 0 (Ω) ∩ D 1,m 0 (Ω) and, by virtue of Proposition 2.3, (u n ) and (Du n ) converge to v 0 and Dv 0 a.e. in Ω, respectively. In turn (see also the proof of Theorem 3.4) it follows for any v ∈ W 1,p 0 (Ω)∩D 1,m 0 (Ω). By combining Theorem 3.2 and Theorem 3.4, setting u 1 n := u n −v 0 and thinking the functions on R N after extension to zero out of Ω, get where (w 1 n ) is a sequence in the dual of W 1,p 0 (Ω) ∩ D 1,m 0 (Ω) with w 1 n → 0 as n → ∞. In turn, it follows that (u 1 n ) is Palais-Smale sequence for φ ∞ at the energy level c − φ(v 0 ). In addition, If it is the case that ̟ = 0, then, according to [11, Lemma I.1], (u 1 n ) converges to zero in L r (R N ) for every r ∈ (p, p * ). Then, one obtains that where the inequality follows by the sign condition (1.9). In turn, testing equation (4.2) with v = u 1 n , by the coercivity and convexity of ξ → L(ξ), M (s, ξ), we have lim sup yielding that (u 1 n ) strongly converges to zero in W 1,p (R N ) ∩ D 1,m (R N ), concluding the proof in this case. If, on the contrary, it holds ̟ > 0, then, there exists an unbounded sequence (y 1 n ) ⊂ R N with B(y 1 n ,1) |u 1 n | p > ̟/2. Whence, let us consider v 1 n := u 1 n (· + y 1 n ), which, up to a subsequence, converges weakly and pointwise to some v 1 ∈ W 1,p (R N ) ∩ D 1,m (R N ), which is nontrivial, due to the inequality B(0,1) |v 1 | p ≥ ̟/2. Notice that, of course, Moreover, since |y 1 n | → ∞ and Ω is an exterior domain, for all ϕ ∈ D(R N ) we have ϕ(·−y 1 n ) ∈ D(Ω) for n ∈ N large enough. Whence, in light of equation (4.2), for every ϕ ∈ D(R N ) we get g(u 1 n )ϕ(· − y 1 n ) = w 1 n , ϕ(· + y 1 n ) .