Multiple solutions for equations of $p(x)$-Laplace type with nonlinear Neumann boundary condition
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In this paper, we are concerned with the nonlinear elliptic equations of the $p(x)$-Laplace type $$ \begin{cases} -\text{div}(a(x,\nabla u))+|u|^{p(x)-2}u=\lambda f(x,u) \quad &\textmd{in} \quad \Omega \\ a(x,\nabla u)\frac {\partial u}{\partial n} = \lambda\theta g(x,u) &\textmd{on} \quad \partial\Omega, \end{cases} $$ which is subject to nonlinear Neumann boundary condition. Here the function $a(x,v)$ is of type $|v|^{p(x)-2}v$ with continuous function $p: \overline{\Omega} \to (1,\infty)$ and the functions $f, g$ satisfy a Carath\'eodory condition. The main purpose of this paper is to establish the existence of at least three solutions for the above problem by applying three critical points theory due to Ricceri. Furthermore, we localize three critical points interval for the given problem as applications of the theorem introduced by Arcoya and Carmona.
Abstract
In this paper, we are concerned with the nonlinear elliptic equations of the $p(x)$-Laplace type $$ \begin{cases} -\text{div}(a(x,\nabla u))+|u|^{p(x)-2}u=\lambda f(x,u) \quad &\textmd{in} \quad \Omega \\ a(x,\nabla u)\frac {\partial u}{\partial n} = \lambda\theta g(x,u) &\textmd{on} \quad \partial\Omega, \end{cases} $$ which is subject to nonlinear Neumann boundary condition. Here the function $a(x,v)$ is of type $|v|^{p(x)-2}v$ with continuous function $p: \overline{\Omega} \to (1,\infty)$ and the functions $f, g$ satisfy a Carath\'eodory condition. The main purpose of this paper is to establish the existence of at least three solutions for the above problem by applying three critical points theory due to Ricceri. Furthermore, we localize three critical points interval for the given problem as applications of the theorem introduced by Arcoya and Carmona.
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