Calculus of Variations and Partial Differential Equations ( IF 2.1 ) Pub Date : 2020-05-13 , DOI: 10.1007/s00526-020-01756-y Tieshan He , Lang He , Meng Zhang
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.
中文翻译:
p -Laplacian的Brézis–Nirenberg型问题:无限多个变号解
在本文中,我们考虑了拟线性临界问题
$$ \ begin {aligned} \ left \ {\ begin {array} {l @ {\ quad} l}-\ Delta _ {p} u = \ lambda \ left | 你\右| ^ {q-2} u + \ left | 你\右| ^ {p ^ {\ star} -2} u&{} \ mathrm {in} \; \ Omega,\\ u = 0&{} \ mathrm {on} \; \ partial \ Omega,\ end {array} \对。\ end {aligned} $$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。