Polyharmonic systems involving critical nonlinearities with sign-changing weight functions

Abstract

This article concerns the existence of multiple solutions of the polyharmonic system involving critical nonlinearities with sign-changing weight functions $$\displaylines{ (-\Delta)^mu = \lambda f(x) |u|^{r-2}u+ \frac{\beta}{\beta+\gamma} h(x) |u|^{\beta-2}u |v|^{\gamma}\quad \text{in }\Omega,\cr (-\Delta)^mv = \mu g(x) |v|^{r-2}v+ \frac{\gamma}{\beta+\gamma} h(x) |u|^{\beta} |v|^{\gamma-2} v \quad \text{in }\Omega, \cr D^ku=D^kv=0\quad \text{for all }|k|\leq m-1\quad \text{on }\partial\Omega, }$$ where $(-\Delta)^m$ denotes the polyharmonic operators, $\Omega$ is a bounded domain in $\mathbb{R}^N$ with smooth boundary $\partial \Omega$, $m\in \mathbb N$, $N\geq {2m+1}$, $11$, $\gamma>1$ satisfying $20$. The functions f, g and $h:\overline{\Omega}\to \mathbb R$ are sign-changing weight functions satisfying f, $g\in L^{\alpha}(\Omega)$ and $h\in L^{\infty}(\Omega)$ respectively. Using the variational methods and Nehari manifold, we prove that the system admits at least two nontrivial solutions with respect to parameter $(\lambda, \mu)\in \mathbb R^2_{+} \setminus \{(0, 0)\}$.

Keywords

• polyharmonic operator system
• sign-changing weight functions
• critical exponent
• nehari manifold
• concave-convex nonlinearities