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

In this paper we consider the quasilinear critical problem where Omega is a bounded domain in RN with smooth boundary, -Delta pu:=div(p-2 del u) is the p-Laplacian, N >= 3,1<p <= q<p star,p star:=Np/N-p, and lambda>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 >= 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.