Multiplicity of solutions for a class of elliptic problem in with Neumann conditions
type: Item
instance of: scholarly article
title: Multiplicity of solutions for a class of elliptic problem in with Neumann conditions
zbMATH Open document ID: 1134.35043
published in: Journal of Differential Equations
publication date: 2005-12-05T00:00:00Z
review text: The paper is concerned with the semilinear elliptic problem \[ -\Delta u + u = Q(x)f(u)~~\text{in } \mathbb{R}^2\setminus\Omega,~~~\frac{\partial u}{\partial \eta} = 0~~\text{on } \partial\Omega, \] where \(\Omega \subset \mathbb{R}^2\) is a bounded smooth domain, \(Q\) is a positive continuous function defined on \({\mathbb R}^2 \setminus \Omega\), with \(Q(x)\to \overline{Q}>0\) as \(|x|\to \infty\), and finally the nonlinear function \(f: \mathbb{R} \to \mathbb{R}\) is continuously differentiable and such that: \((f_0)\) there exists \(\alpha_0>0\) such that \[ \lim_{|s|\to\infty} \frac{|f(s)|}{\exp(\alpha s^2)} = 0\quad\forall~\alpha>\alpha_0,\qquad\lim_{|s|\to\infty} \frac{|f(s)|}{\exp(\alpha s^2)} = +\infty\quad\forall~\alpha<\alpha_0, \] \((f_1)\) \(|f(s)| \leq C \exp(4\pi s^2)\) for all \(s \in \mathbb{R}\), \((f_2)\) there exists \(\theta>2\) such that \(0 < \theta F(s) \leq sf(s)\) for all \(s \in \mathbb{R}\), \((f_3)\) there exists \(q>1\) such that \( \limsup_{s \to 0} \frac{|f(s)|}{|s|^q} < \infty, \) \((f_4)\) the function \(s \mapsto f(s)/s\) is increasing in \((0,\infty)\), \((f_5)\) there exists \(p>2\) and \(C_p>0\) such that \[ f(s) \geq C_ps^{p-1}~~~\forall~s \in [0,\infty), \] where \(C_p\) is a suitable number which depends on \(\Omega,p,\theta\) and \(\bar Q\). The author, motivated by a paper of \textit{D. M. Cao} [Commun. Partial Differ. Equations 18, No. 3--4, 687--700 (1993; Zbl 0803.35049)] proves the existence of a positive and a nodal solution. The positive solution is a ground state solution and it is obtained under the additional condition (\(Q_2\)) there exists \( R_0>0\) and \(m>2\), such that \[ Q(x) \geq \bar Q - C\text{e}^{-m|x|}~~~|x|>R_0. \] In the second result the author obtained a nodal solution under the additional conditions (\(f_6\)) there exists \(\sigma \geq 2\) such that \(f'(s)s^2 - f(s)s \geq C|s|^{\sigma}\) for all \(s\in\mathbb{R}\), (\(f_7\)) \(|f'(s)s| \leq C\exp(4\pi s^2)\) for all \(s \in\mathbb{R}\), (\(Q_3\)…
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 2235678
zbMATH Keywords: elliptic problem
zbMATH Keywords: critical growth
zbMATH Keywords: Trudinger-Moser inequality
author: Claudianor O. Alves
MaRDI profile type: Publication
full work available at URL: j.jde.2004.11.010
OpenAlex ID: W2060670484
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DOI: 10.1016/J.JDE.2004.11.010
description: scientific article