L∞$L^\infty$ compactness of solutions of quasilinear problems and applications

Abstract

For a (not necessarily smooth) bounded domain Ω$\Omega$ of RN$\mathbb {R}^N$ , N⩾2$N \geqslant 2$ and a Carathéodory vector‐valued function a:Ω×RN→RN$a:\Omega \times \mathbb {R}^N \rightarrow \mathbb {R}^N$ , we study the compactness of the inverse of the Leray–Lions operator A(u)=−div(a(x,∇u))$A(u)=-\text{div}(a(x, \nabla u))$ , u∈W01,p(Ω)$u\in W_0^{1, p}(\Omega)$ , 1max{N/p,(p∗/2)′}$\sigma >\max \lbrace N/p, (p^{*}/2)^{^{\prime }} \rbrace $ . Also, for σ>p∗:=(p∗)′$\sigma >p_*:=(p^*)^{\prime }$ , the operators A−1:Lσ(Ω)→W01,p(Ω)$A^{-1}:L^\sigma (\Omega) \rightarrow W_0^{1,p}(\Omega)$ and A−1:Lσ(Ω)→Lq(Ω)$A^{-1}:L^\sigma (\Omega) \rightarrow L^q(\Omega)$ are compact for every 1⩽q<[(p−1)σ∗]∗$1 \leqslant q <[(p-1)\sigma ^*]^*$ . In contrast, it is also established that A−1:Lp∗(Ω)⟶Lp∗(Ω)$A^{-1}: L^{p_*}(\Omega) \longrightarrow L^{p^*}(\Omega)$ and A−1:Lp∗(Ω)→W01,p(Ω)$A^{-1}: L^{p_*}(\Omega) \rightarrow W_0^{1, p}(\Omega)$ are not compact. The compactness of the set of solutions for the more general operator A(u)=−div(a(x,u,∇u))$A(u)=-\text{div}(a(x, u, \nabla u))$ is also studied. The proofs of the main results are based on an appropriate adaptation of Stampacchia's technique, which provides a more elementary approach than the usual regularity theory up to the boundary, that cannot be employed for nonsmooth domains. As an application, we improve previous results on the existence and multiplicity of solutions for a class of problems involving the p$p$ ‐Laplacian operator under a local Landesman–Lazer condition for arbitrary bounded domains.