Nonlinear eigenvalue problem involving the p(x)-Laplacian operator: existence and uniqueness results

Abstract

In this dissertation, we analyze a nonlinear eigenvalue elliptic problem, inspired by models from mathematical physics, particularly in nonlinear quantum mechanics and field theory. More specifically, we examine the p(x)-Laplacian equation of the form:    −∆p(x)

u(x)

= λ ∥u(x)∥ p(x)−q(x) q(x)

u(x)

q(x)−2 u(x), x ∈ Ω, u(x) = 0, x ∈ ∂Ω. (1) The given equation is defined on an open and regular domain Ω, with λ ∈ R representing a spectral parameter and u being the corresponding eigenfunction, where: 1 < p− ≤ p(x) ≤ p+ < +∞, 1 < q− ≤ q(x) ≤ q+ < p∗ (x) with p ∗ (x) =    n p(x) n − p(x) , p+ < n, ∞, p+ ≥ n. (2) Such that: r− = ess inf x∈Ω r(x), r+ = ess sup x∈Ω r(x). The operator p(x)-Laplacian is defined by: ∆p(x)

u(x)

= div |∇u(x)| p(x)−2 ∇u(x)

, and the notation ∥u∥q(x) is the norm of u in the space L q(x) (Ω). We look into the right variational methods for this case, along with whether solutions exist and how many there are, using concepts from variable exponent Lebesgue and Sobolev space theories, as well as nonlinear spectral theory.