Bifurcation in two parameters for a quasilinear Schrödinger equation

Abstract

This paper deals with existence and multiplicity of positive solutions for the quasilinear Schrödinger equation \begin{align*} \left\{ \begin{aligned} -\Delta u - \lambda m(x) u \Delta(u^2) &= f(\mu, x, u)&&\text{in } \Omega, \\ u &= 0&&\text{on } \partial\Omega. \end{aligned}\right. \end{align*} where $\Omega$ is a bounded open domain in $\mathbb{R}^N$ with smooth boundary and $m$ is a bounded non negative continuous function. Under suitable assumptions on the asymptotically linear $f$, we use bifurcation theory to analyze the set of positive solutions.

Keywords

• bifurcation theory
• schrödinger equation