Nontrivial Solution for a Nonlocal Elliptic Transmission Problem in Variable Exponent Sobolev Spaces

On a nonlinear eigenvalue problem in Sobolev spaces with variable exponent

Abstract. We consider a class of nonlinear Dirichlet problems involving the p(x)–Laplace operator. Our framework is based on the theory of Sobolev spaces with variable exponent and we establish the existence of a weak solution in such a space. The proof relies on the Mountain Pass Theorem.

Nonlinear eigenvalue problems in Sobolev spaces with variable exponent

Abstract. We study the boundary value problem −div((|∇u|1 + |∇u|2)∇u) = f(x, u) in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in R . We focus on the cases when f±(x, u) = ±(−λ|u| u+ |u|u), where m(x) := max{p1(x), p2(x)} < q(x) < N ·m(x) N−m(x) for any x ∈ Ω. In the first case we show the existence of infinitely many weak solutions for any λ > 0. In the second case we prove that if λ is...

On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent

We consider the nonlinear eigenvalue problem −div ( |∇u|∇u ) = λ|u|u in Ω, u = 0 on ∂Ω, where Ω is a bounded open set in R with smooth boundary and p, q are continuous functions on Ω such that 1 < infΩ q < infΩ p < supΩ q, supΩ p < N , and q(x) < Np(x)/ (N − p(x)) for all x ∈ Ω. The main result of this paper establishes that any λ > 0 sufficiently small is an eigenvalue of the above nonhomogene...

Positive solutions for a nonlinear nonlocal elliptic transmission problem

K e y w o r d s E l l i p t i c system, Transmission problem, Positive solution. 1. I N T R O D U C T I O N Let 12 be a smooth bounded domain of R n, n > 2, and let ~1 c gl be a subdomain with smooth boundary ~ satisfying ~1 C i2. Writing F = c9~ and gl2 = ~\~l l we have ~ = ~1 tJ 122 and 0i'~2 = ~, tJ F. We are concerned with the existence of positive solutions to the following system of nonli...

Existence of Nontrivial Solution for a Nonlocal Elliptic Equation with Nonlinear Boundary Condition