The Brézis–Nirenberg type problem for the p-Laplacian: infinitely many sign-changing solutions

Abstract

In this paper we consider the quasilinear critical problem $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\Delta _{p}u=\lambda \left| u\right| ^{q-2}u+\left| u\right| ^{p^{\star }-2}u &{} \mathrm {in}\;\Omega ,\\ u=0 &{} \mathrm {on}\;\partial \Omega , \end{array}\right. \end{aligned}$$where $$\Omega $$ is a bounded domain in $${\mathbb {R}}^{N}$$ with smooth boundary, $$-\Delta _{p}u:=\mathrm {div}(\left| \nabla u\right| ^{p-2}\nabla u)$$ is the p-Laplacian, $$N\ge 3,1 0$$ is a parameter. We investigate the multiplicity of sign-changing solutions to the problem and find the phenomenon depending on the positive solutions. Precisely, we show that the problem admits infinitely many pairs of sign-changing solutions when a positive solution exists. These results complete those obtained in Schechter and Zou (On the Brezis–Nirenberg problem, Arch Rational Mech Anal 197:337–356, 2010) for the cases $$p=q=2$$ and $$N\ge 7$$, and in Azorero and Peral (Existence and nonuniqueness for the p-Laplacian: nonlinear eigenvalues, Commun Partial Differ Equ 12:1389–1430, 1987) for the case of one positive solution. Our approach is based on variational methods combining upper-lower solutions and truncation techniques, and flow invariance arguments.