General class of optimal Sobolev inequalities and nonlinear scalar field equations

Jarosław Mederski

Published 2018-12-30Version 1

We find a class of optimal Sobolev inequalities $$\Big(\int_{\mathbb{R}^N}|\nabla u|^2\, dx\Big)^{\frac{N}{N-2}}\geq C_{N,G}\int_{\mathbb{R}^N}G(u)\, dx, \quad u\in\mathcal{D}^{1,2}(\mathbb{R}^N), N\geq 3,$$ where the nonlinear function $G:\mathbb{R}\to\mathbb{R}$ satisfies general assumptions in the spirit of the fundamental works of Berestycki and Lions involving zero, positive as well as infinite mass cases. We show that any minimizer is radial up to a translation, moreover, up to a dilation, it is a least energy solution of the nonlinear scalar field equation $$-\Delta u = g(u)\quad \hbox{in }\mathbb{R}^N,\quad\hbox{with }g=G'.$$ In particular, if $G(u)=u^2\log |u|$, then the sharp constant is $C_{N,G}:=2^*(\frac{N}{2})^{2^*}e^{\frac{2(N-1)}{N-2}}(\pi)^{\frac{N}{N-2}}$ and $u_\lambda(x)=e^{\frac{N-1}{2}-\frac{\lambda^2}{2}|x|^2}$ with $\lambda>0$ constitutes the whole family of minimizers up to translations. The above optimal inequality provides a simple proof of the classical logarithmic Sobolev inequality. Moreover, if $N\geq 4$, then there is at least one nonradial solution and if, in addition, $N\neq 5$, then there are infinitely many nonradial solutions of the nonlinear scalar field equation. The energy functional associated with the problem may be infinite on $\mathcal{D}^{1,2}(\mathbb{R}^N)$ and is not Fr\'echet differentiable in its domain. We present a variational approach to this problem based on a new variant of Lions' lemma in $\mathcal{D}^{1,2}(\mathbb{R}^N)$.