Multiple solutions for nonlocal non-homogeneous Neumann problems in Orlicz-Sobolev spaces

  • Nguyen Thanh Chung Quang Binh University

Abstract

Using variational method, we study the existence of multiple solutions to the following nonlocal nonhomogeneous Neumann problem
\begin{equation*}
\begin{cases}
- M\left(\rho(u)\right)\left(\mathrm{div}(a(|\nabla u|)\nabla u)-a(|u|)u\right) = \lambda
f(x, u)+\mu g(x,u) \quad \text{ in } \Omega,\\
\frac{\partial u}{\partial \nu} = 0 \quad \text{ on } \partial\Omega,
\end{cases}
\end{equation*}
where $\Omega \subset \R^N$, $N \geq 3$ is a smooth bounded domain, $\nu$ is the outward unit normal to $\partial\Omega$, $\rho(u)= \int_\Omega \left(\Phi(|\nabla u|)+\Phi(|u|) \right)\, dx$, $\lambda,\mu$ are positive parameters, $f,g: \Omega \times \R \to \R$ are Carath\'{e}odory functions, $M: \R^+_0 \to \R$ is a continuous and nondecreasing function that may be degenerate at zero.

Published
2016-08-28
Section
Articles