Abstract
On an eigenvalue problem with variable exponents and sign-changing potential
Highlights
Let Ω ⊂ RN(N ≥ 3) be a bounded domain with smooth boundary ∂Ω
In this note we study the following nonlinear eigenvalue problem:
The interest in analyzing this kind of problems is motivated by some recent advances in the study of eigenvalue problems involving non-homogeneous operators in the divergence form
Summary
Let Ω ⊂ RN(N ≥ 3) be a bounded domain with smooth boundary ∂Ω. We assume that the function a : (0, ∞) → R is such that the mapping φ : R → R defined by φ(t) = a(|t|)t, for t = 0, 0, for t = 0, is an odd, increasing homeomorphism from R onto R. In order to go further we introduce the functional space setting where problem (P) will be discussed. We observe that Φ∗ is an N-function and the following Young’s inequality holds true: st ≤ Φ(s) + Φ∗(t), ∀s, t ≥ 0. We denote by W01LΦ(Ω) the corresponding Orlicz–Sobolev space for problem (P), equipped with the norm u = ∇u Φ
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