D s,2 ( R N ) versus C ( R N ) local minimizers

Abstract

Let $\mathit{s}\in \mathrm{(}0\mathrm{,}1\mathrm{)}$ , $\mathit{N}\mathrm{>}2\mathit{s}$ and $$\begin{array}{c}{\mathcal{D}}^{\mathit{s}\mathrm{,}2}\mathrm{\left(}{\mathbb{R}}^{\mathit{N}}\mathrm{\right)}:=\{\mathit{u}\in {\mathit{L}}^{2_{\mathit{s}}^{\ast }}\mathrm{(}{\mathbb{R}}^{\mathit{N}}\mathrm{)}:\Vert \mathit{u}{\Vert }_{{\mathcal{D}}^{\mathit{s}\mathrm{,}2}\mathrm{(}{\mathbb{R}}^{\mathit{N}}\mathrm{)}}:={\mathrm{\left(}{\displaystyle \frac{{\mathit{C}}_{\mathit{N}\mathrm{,}\mathit{s}}}{2}}{\iint }_{{\mathbb{R}}^{2\mathit{N}}}{\displaystyle \frac{|\mathit{u}\mathrm{(}\mathit{x}\mathrm{)}-\mathit{u}\mathrm{(}\mathit{y}\mathrm{)}{|}^2}{|\mathit{x}-\mathit{y}{|}^{\mathit{N}+2\mathit{s}}}}\mathit{d}\mathit{x}\mathit{d}\mathit{y}\mathrm{\right)}}^{\frac{1}{2}}\mathrm{<}\infty \}\mathrm{,}\end{array}$$ where $2_{\mathit{s}}^{\ast }:=\frac{2\mathit{N}}{\mathit{N}-2\mathit{s}}$ is the fractional critical exponent and ${\mathit{C}}_{\mathit{N}\mathrm{,}\mathit{s}}$ is a positive constant. We consider functionals $\mathit{J}:{\mathcal{D}}^{\mathit{s}\mathrm{,}2}\mathrm{(}{\mathbb{R}}^{\mathit{N}}\mathrm{)}\to \mathbb{R}$ of the type $$\begin{array}{l}\hfill \mathit{J}\mathrm{(}\mathit{u}\mathrm{)}:={\displaystyle \frac{1}{2}}\Vert \mathit{u}{\Vert }_{{\mathcal{D}}^{\mathit{s}\mathrm{,}2}\mathrm{(}{\mathbb{R}}^{\mathit{N}}\mathrm{)}}^2-{{\displaystyle \int }}_{{\mathbb{R}}^{\mathit{N}}}\mathit{b}\mathrm{(}\mathit{x}\mathrm{)}\mathit{G}\mathrm{(}\mathit{u}\mathrm{)}\mathit{d}\mathit{x}\mathrm{,}\end{array}$$ where $\mathit{G}\mathrm{(}\mathit{t}\mathrm{)}:={\int }_0^{\mathit{t}}\mathit{g}\mathrm{(}\mathit{\tau }\mathrm{)}\mathit{d}\mathit{\tau }$ , $\mathit{g}:\mathbb{R}\to \mathbb{R}$ is a continuous function with subcritical growth at infinity, and $\mathit{b}:{\mathbb{R}}^{\mathit{N}}\to \mathbb{R}$ is a suitable weight function. We prove that a local minimizer of J in the topology of the subspace $$\begin{array}{c}{\mathcal{V}}_{\mathit{s}}:=\{\mathit{u}\in {\mathcal{D}}^{\mathit{s}\mathrm{,}2}\mathrm{(}{\mathbb{R}}^{\mathit{N}}\mathrm{)}:\mathit{u}\in \mathit{C}\mathrm{(}{\mathbb{R}}^{\mathit{N}}\mathrm{\left)}\text{ with }\underset{\mathit{x}\in {\mathbb{R}}^{\mathit{N}}}{sup}\mathrm{\right(}1+|\mathit{x}{|}^{\mathit{N}-2\mathit{s}}\mathrm{)}|\mathit{u}\mathrm{(}\mathit{x}\mathrm{)}|\mathrm{<}\infty \}\end{array}$$ must be a local minimizer of J in the ${\mathcal{D}}^{\mathit{s}\mathrm{,}2}\mathrm{(}{\mathbb{R}}^{\mathit{N}}\mathrm{)}$ -topology.