On a Brezis–Nirenberg type quasilinear problem

Abstract

In this paper, we consider the following quasilinear equation with critical exponent, a quasilinear version of the classical Brezis–Nirenberg problem: $$\begin{aligned} \left\{ \begin{array}{rlll} -\displaystyle \sum _{i, j=1}^ND_j(a_{ij}(u)D_i u)+\frac{1}{2}\displaystyle \sum _{i, j=1}^Na'_{ij}(u)D_i uD_j u &{}= au+|u|^{2^*-2}u &{} \quad \hbox { in } \Omega \\ u&{}=0 &{} \quad \hbox { on } \partial \Omega ,\\ \end{array}\right. \quad {(P)} \end{aligned}$$ where \( 2^*=\frac{2N}{N-2}\) is the Sobolev critical exponent for the embedding of \(H_0^1(\Omega )\) into \(L^p(\Omega )\), \(\Omega \subset \mathbb {R}^N\) is an open-bounded domain with smooth boundary, \(a>0\) is a constant. We prove that if \(N\ge 7,\) the problem (P) admits an unbounded sequence of solutions. These solutions are obtained from subcritical approximations. This is done by analyzing the asymptotic behavior and the concentration compactness of the approximation solutions for the subcritical problems. The existence of infinitely many solutions to the subcritical problems is obtained via a p-Laplacian regularization process.