𝒟1,2(ℝN) versus C(ℝN) local minimizer on manifolds and multiple solutions for zero-mass equations in ℝN

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}.