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 \(\partial \Omega\), \(m\in \mathbb N\), \(N\geq {2m+1}\), \(1<r<2\) and \(\beta>1\), \(\gamma>1\) satisfying \(2<\beta+\gamma\leq 2_{m}^{*}\) with \(2_{m}^{*}=\frac{2N}{N-2m}\) as a critical Sobolev exponent and \(\lambda\), \(\mu>0\). 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 \((\lambda, \mu)\in \mathbb R^2_{+} \setminus \{(0, 0)\}\). For more information see https://ejde.math.txstate.edu/Volumes/2020/119/abstr.html