Affiliation:
1. Institut für Mathematik, Martin-Luther-Universität Halle-Wittenberg, Halle, Germany
2. Department of Mathematical Sciences, University of Nevada Las Vegas, Box 454020, Las Vegas, USA
Abstract
AbstractWe consider functionals of the formJ(u)=\frac{1}{2}\int_{{\mathbb{R}}^{N}}|\nabla u|^{2}-\int_{{\mathbb{R}}^{N}}b%
(x)G(u)on a {C^{1}}-submanifold M of
{\mathcal{D}^{1,2}({\mathbb{R}}^{N})}, {N\geq 3}, where G is the primitive of some “zero-mass” nonlinearity g (i.e., {g^{\prime}(0)=0}), and the weight function {b:{\mathbb{R}}^{N}\to{\mathbb{R}}} is merely supposed to belong to {L^{1}({\mathbb{R}}^{N})\cap L^{\frac{2^{*}}{2^{*}-p}}({\mathbb{R}}^{N})} for some {2<p<2^{*}}, and to possess a certain decay behavior. Let V be the subspace of {\mathcal{D}^{1,2}({\mathbb{R}}^{N})} given by {V:=\{v\in\mathcal{D}^{1,2}({\mathbb{R}}^{N}):v\in C({\mathbb{R}}^{N})\mbox{ %
with }\sup_{x\in{\mathbb{R}}^{N}}(1+|x|^{N-2})|v(x)|<\infty\}}. We prove that a local minimizer of the constrained functional {J|_{M}} with respect to the V-topology must be a local minimizer with respect to the “bigger” {\mathcal{D}^{1,2}({\mathbb{R}}^{N})}-topology. This result allows us to prove the existence of multiple nontrivial solutions of the zero-mass equation {-\Delta u=b(x)g(u)} in {{\mathbb{R}}^{N}}, where {g:R\to{\mathbb{R}}} is a subcritical nonlinearity, which is superlinear at zero and at {\infty}.
Subject
Applied Mathematics,Analysis
Cited by
4 articles.
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