Existence and regularity of multiple solutions for infinitely degenerate nonlinear elliptic equations with singular potential
type: Item
instance of: scholarly article
title: Existence and regularity of multiple solutions for infinitely degenerate nonlinear elliptic equations with singular potential
zbMATH Open document ID: 1303.35035
author: Hua Chen
author: Peng Luo
author: Shuying Tian
published in: Journal of Differential Equations
publication date: 2014-09-12T00:00:00Z
review text: This paper is devoted to the study of the existence and regularity of solution to the following semilinear infinitely degenerate elliptic equation \[ \begin{cases} -\Delta_X u-\varepsilon V_n u= au \log |u|+b u+ g(x,u) &\text{in } \Omega, \\ u=0&\text{on } \partial\Omega, \end{cases}\eqno{(1.1)} \] where vector fields \(X=(\partial_{x_1}, \dots , \partial_{x_{n-1}}, \varphi(x')\partial_{x_n})=(X_1,\dots, X_n)\) defined on an open domain \(\widetilde{\Omega}\subset\mathbb R^n\) for \(n \geq 3\), \(\Omega\) is a bounded open subset in \(\widetilde{\Omega}\) which contains the origin, \(a>0, b>0; \varphi(x')\) is a non-negative \(C^\infty\)-smooth function in \(x'=(x_1, x_2, \dots , x_{n-1})\) and for \(\Gamma \subset \widetilde{\Omega}, \partial^\alpha_{x'} \varphi(x')|_{(x',x_n)\in \Gamma} =0\) for any \(|\alpha| \geq 0; g(x, u)\) is a Carathéodory function with primitive \(G(x, u) =\int_0^u g(x,v)\, dv\), which satisfies the following conditions: {\parindent=8mm \begin{itemize} \item[(\(\mathit{f}_1\))] \(g(x, u) =-g(x,-u)\), \item[(\(\mathit{f}_2\))] there exist \(q\in (1, 2)\), and \(c>0\), such that \(|g(x, u)| \leq c(1 +|u|^{q-1})\), \item[(\(\mathit{f}_3\))] if \(u \in L^2(\Omega )\) and \(u\not=0\), then \(\lim_{t \to 0^+}\frac{\int_\Omega G(x,tu)\, dx}{t^2 \log t}=-\infty \), \item[(\(\mathit{f}_4\))] there exists \(\varepsilon_0>0\), such that for \(0 \leq u \leq\varepsilon_0\), we have \(g(x,u)+au \log |u| +bu \geq 0\), \item[(\(\mathit{f}_5\))] \(g(x, u)\) is \(C^\infty\) in \(x\), and \(C^\infty\) in \(u\) except \(u =0\). \end{itemize}} The authors suppose that the vector fields \(X\) satisfies the following logarithmic regularity estimates \[ \|(\log\Lambda)^su \|^2_{L^2(\Omega)} \leq C_0 \Bigl[ \int_\Omega | Xu |^2 \,dx + \|u\|^2_{L^2(\Omega)}\Bigr] \eqno(1.2) \] for all \(u \in C^\infty_0(\widetilde{\Omega})\), where \(\Lambda =(e^2+|D|^2)^{\frac{1}{2}}=\langle D\rangle\). The potential term \(V_n(x) \geq 0\) satisfies the following Hardy's inequality…
reviewed by: Petr Tomiczek
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35D30
Mathematics Subject Classification ID: 49J35
zbMATH DE Number: 6343733
zbMATH Keywords: infinitely degenerate elliptic equations
zbMATH Keywords: logarithmic Sobolev inequality
zbMATH Keywords: Hardy's inequality
zbMATH Keywords: singular potential
MaRDI profile type: Publication
full work available at URL: j.jde.2014.06.014
OpenAlex ID: W1988109314
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DOI: 10.1016/J.JDE.2014.06.014
description: scientific article; zbMATH DE number 6343733