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

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

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The authors thank the very knowledgeable referee very much for his/her valuable comments and helpful suggestions.

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Correspondence to Tieshan He.

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Communicated by M. Struwe.

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Supported by Guangdong Basic and Applied Basic Research Foundation (No. 2020A1515010459) and by Social science-Philosophy plan Project of Guangzhou (No. 2019GZGJ125).

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He, T., He, L. & Zhang, M. The Brézis–Nirenberg type problem for the p-Laplacian: infinitely many sign-changing solutions. Calc. Var. 59, 98 (2020). https://doi.org/10.1007/s00526-020-01756-y

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