Multiple positive solutions to critical p-Laplacian equations with vanishing potential,Zeitschrift für angewandte Mathematik und Physik

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Multiple positive solutions to critical p-Laplacian equations with vanishing potential
Zeitschrift für angewandte Mathematik und Physik ( IF 1.7 ) Pub Date : 2021-08-04 , DOI: 10.1007/s00033-021-01598-4
Lun Guo ₁ , Qi Li _{2,

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Affiliation

This paper deals with the following p-Laplacian equation

$$\begin{aligned} -\varepsilon ^{p}\Delta _{p}u+V(x)|u|^{p-2}u=|u|^{p^{*}-2}u,\quad u\in D^{1,p}({\mathbb {R}}^N), \end{aligned}$$

where $p\in (1,N)$, p-Laplacian operator $\Delta _{p}{:}{=}$div$(|\nabla u|^{p-2}\nabla u) $, $p^{*}=Np/(N-p)$, $\varepsilon $ is a positive parameter, $V(x)\in L^{{N}/{p}}({\mathbb {R}}^N)\cap L^{\infty }_{loc}({\mathbb {R}}^N)$ and V(x) is assumed to be zero in some region of ${\mathbb {R}}^N$, which means it is of the vanishing potential case. In virtue of Ljusternik–Schnirelman theory of critical points, we succeed in proving the multiplicity of positive solutions. This result generalizes the result for semilinear Schrödinger equation by Chabrowski and Yang (Port. Math. 57 (2000), 273–284) to p-Laplacian equation.

中文翻译：

具有消失势的临界 p-Laplacian 方程的多重正解

本文涉及以下p-拉普拉斯方程

$$\begin{aligned} -\varepsilon ^{p}\Delta _{p}u+V(x)|u|^{p-2}u=|u|^{p^{*}-2} u,\quad u\in D^{1,p}({\mathbb {R}}^N), \end{aligned}$$

其中$p\in (1,N)$ , p -拉普拉斯算子$\Delta _{p}{:}{=}$ div $(|\nabla u|^{p-2}\ nabla u) $ , $p^{*}=Np/(Np)$ , $\varepsilon $是一个正参数，$V(x)\in L^{{N}/{p }}({\mathbb {R}}^N)\cap L^{\infty }_{loc}({\mathbb {R}}^N)$并且V ( x ) 在某些情况下被假定为零的区域\（{\ mathbb {R}} ^ N \），这意味着它是消失的潜在情况。凭借 Ljusternik-Schnirelman 临界点理论，我们成功地证明了正解的多样性。该结果将 Chabrowski 和 Yang (Port. Math. 57 (2000), 273–284) 的半线性薛定谔方程的结果推广到p-拉普拉斯方程。

更新日期：2021-08-09

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