Strong comparison principle and multiplicity results for Finsler p-Laplace equations

Abstract

In this paper, we consider the family of problems \begin{equation*} \begin{cases} - \Delta_{H, p} u = f_{\lambda}(x, u) & \text{in } \Omega, \\ u > 0 & \text{in } \Omega, \\ u = 0 & \text{on } \partial\Omega, \end{cases} \end{equation*} where $\Omega$ is a smooth bounded domain in ${\mathbb{R}}^N$, $N\ge 2$ , $\lambda$ is a real parameter and the anisotropic Finsler $p$-Laplacian operator $\Delta_{H, p}$ with $1< p< N$, is defined as $\Delta_{H, p} u: = \text{div}(H(\nabla u)^{p-1} \nabla_{\eta} H(\nabla u)).$ The non-linear term $f_\lambda \colon \Omega \times \mathbb{R} \to \mathbb{R} $ is the sum of a sublinear and a superlinear term. We establish the strong comparison principle and extend the well-known Brezis-Nirenberg result concerning local minimization in $C_0^1$ and $W_0^{1,p}$ to the framework of the Finsler $p$-Laplace operator. Under various growth assumptions on the non-linear term $f_\lambda (x, s)$ and the real parameter $\lambda$, we establish the existence, non-existence and multiplicity of solutions.