In this paper we consider the quasilinear critical problem

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, \(N\ge 3,1<p\le q<p^{\star },p^{\star }:=Np/N-p\), 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 Brézis–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.

在本文中，我们考虑了拟线性临界问题

其中\（\ Omega \）是\（{\ mathbb {R}} ^ {N} \）中具有光滑边界的有界域，\（-\ Delta _ {p} u：= \ mathrm {div}（\ left | \ nabla u \ right | ^ {p-2} \ nabla u）\）是p -Laplacian，\（N \ ge 3,1 <p \ le q <p ^ {\ star}，p ^ { \ star}：= Np / Np \），而\（\ lambda> 0 \）是一个参数。我们研究了该问题的多种符号转换解决方案，并根据正解找到了现象。精确地，我们证明了当存在一个正解时，该问题可以无限次接纳成对的变号解。这些结果完成了在Schechter和Zou（关于Brézis-Nirenberg问题，Arch Rational Mech Anal 197：337-356，2010）中获得的结果\（p = q = 2 \）和\（N \ ge 7 \），以及在Azorero和Peral中（p -Laplacian的存在性和非唯一性：非线性特征值，Commun Partial Differ Equ Equ 12：1389–1430，1987）一个积极的解决方案。我们的方法基于结合了上下解和截断技术的变分方法，以及流量不变性参数。