Infinitely many bound states for some nonlinear scalar field equations
type: Item
instance of: scholarly article
title: Infinitely many bound states for some nonlinear scalar field equations
zbMATH Open document ID: 1078.35113
published in: Calculus of Variations and Partial Differential Equations
publication date: 2005-05-12T00:00:00Z
review text: Existence of infinitely many \(H^1(\mathbb{R}^N)\) solutions to the equation \[ -\Delta u+ a(x)u= |u|^{p-2}u,\quad N\geq 2,\;p> 2,\;p<{2N\over N-2},\;\text{ when }N> 2,\tag{1} \] is proved without symmetry assumptions on \(a(x)\). The solutions are critical points of the energy functional \[ I(u)= {1\over 2} \int_{\mathbb{R}^N} (|\nabla u|^2+ a(x) u^2)\,dx- {1\over p} \int_{\mathbb{R}^N} |u|^p\,dx. \] Variational methods are useful to study (1) when considered on a bounded domain in \(\mathbb{R}^N\). But on \(\mathbb{R}^N\), without assuming symmetry of \(a(x)\), variational methods can not be used. To overcome this difficulty, the authors assume \(a\in C^1(\mathbb{R}^N)\), \(\liminf_{|x|\to\infty}a(x)= a_\infty> 0\) and \[ {\partial a\over\partial\xi}(x) e^{\alpha|x|}\to\infty,\quad |\nabla_{\tau_x}a(x)|\leq \overline c{\partial a\over\partial \xi}(x),\quad |x|>\overline c>1,\quad \xi= {x\over|x|}. \] Then the authors consider a sequence of balls \(B_{\rho_n}= \{|x|< \rho_n\}\) in \(\mathbb{R}^N\) and the sequence \(\{u_n\}\) of solutions of (1) on \(B_{\rho_n}\) with the boundary condition \(u_n= 0\) on \(\partial B_{\rho_n}\). Precompactness of this sequence is shown by proving a local Pokhozhaev type inequality (Lemma 4.2 and 4.3). Then taking a suitable subsequence of \(\{u_n\}\), existence of infinitely many solutions to (1) is proved (Th.1.1. Proof is given in \S5). Useful facts and preliminary results are summarized in \S2. \S3 gives preliminary results to \S4. The authors remark that the same result holds on \(\mathbb{R}^N\setminus\Omega\), where \(\Omega\) is a bounded smooth open set in \(\mathbb{R}^N\).
reviewed by: Akira Asada
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 81Q05
zbMATH DE Number: 2167753
zbMATH Keywords: Sobolev space
zbMATH Keywords: precompactness of a sequence
zbMATH Keywords: Pokhozhaev type inequality
author: Giovanna Cerami
author: Giuseppe Devillanova
author: Sergio Solimini
MaRDI profile type: Publication
full work available at URL: s00526-004-0293-6
OpenAlex ID: W2044584297
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DOI: 10.1007/S00526-004-0293-6
description: scientific article; zbMATH DE number 2167753