A nonlinear eigenvalue problem with <inline-formula><tex-math id="M1">\begin{document}$ p(x) $\end{document}</tex-math></inline-formula>-growth and generalized Robin boundary value condition

Abstract

We are concerned with the study of the following nonlinear eigenvalue problem with Robin boundary condition    −div (a(x, ∇u)) = λb(x, u) in Ω ∂A ∂n + β(x)c(x, u) = 0 on ∂Ω. The abstract setting involves Sobolev spaces with variable exponent. The main result of the present paper establishes a sufficient condition for the existence of an unbounded sequence of eigenvalues. Our arguments strongly rely on the Lusternik-Schnirelmann principle. Finally, we focus to the following particular case, which is a p(x)-Laplacian problem with several variable exponents:    −div (a 0 (x)|∇u| p(x)−2 ∇u) = λb 0 (x)|u| q(x)−2 u in Ω |∇u| p(x)−2 ∂u ∂n + β(x)|u| r(x)−2 u = 0 on ∂Ω. Combining variational arguments, we establish several properties of the eigenvalues family of this nonhomogeneous Robin problem.