Existence and multiplicity of solutions for quasilinear nonhomogeneous problems: An Orlicz–Sobolev space setting

Abstract

We study the boundary value problem − div(log(1 + |∇u| q )|∇u| p−2 ∇u) = f (u) in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in R N with smooth boundary. We distinguish the cases where either f (u) = −λ|u| p−2 u + |u| r−2 u or f (u) = λ|u| p−2 u − |u| r−2 u, with p, q > 1, p + q < min{N, r}, and r < (Np − N + p)/(N − p). In the first case we show the existence of infinitely many weak solutions for any λ > 0. In the second case we prove the existence of a nontrivial weak solution if λ is sufficiently large. Our approach relies on adequate variational methods in Orlicz-Sobolev spaces.

Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent

“…Many properties of Sobolev spaces have been extended to Orlicz-Sobolev spaces, mainly by Donaldson and Trudinger [9], and O'Neill [26] (see also Adams [2] for an excellent account of those works). The spaces L p(·) (Ω) and W m,p(·) (Ω) were thoroughly studied in the monograph by Musielak [25] and the papers by Edmunds et al [10][11][12], Kovacik and Rákosník [21], Mihȃilescu and Rȃdulescu [22][23][24], and Samko and Vakulov [34]. Variable Sobolev spaces have been used in the last decades to model various phenomena.…”

Nonhomogeneous boundary value problems in Orlicz–Sobolev spaces

“…It follows that {u n } is bounded in E. Thus, there exists u 0 ∈ E such that, up to a subsequence, {u n } converges weakly to u 0 in E. Since E is compactly embedded in L p (Ω) and L r (Ω) it follows that {u n } converges strongly to u 0 in L p (Ω) and L r (Ω We refer to [10] for further results in the study of quasilinear nonhomogeneous problems in Orlicz-Sobolev spaces.…”

Eigenvalue problems for anisotropic equations involving a potential on Orlicz-Sobolev type spaces

“…Lemma 5.1 ensures us that the functional T λ is coercive on E. On the other hand, using Lemma 1 in [19], similar arguments as those used in the proof of Theorem 2 in [20] lead us to the fact that T λ is weakly lower semicontinuous, as well. So, we have the necessary data to apply Theorem 1.2 in [32] to obtain the existence of an element u ∈ E, global minimizer of T λ and, consequently, the weak solution of problem (1.1).…”