General class of optimal Sobolev inequalities and nonlinear scalar field equations

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 $G:\mathbb{R}\to\mathbb{R}$ of class $\mathcal{C}^1$ satisfies general growth assumptions in the spirit of the fundamental works of Berestycki and Lions. We admit, however, a wider class of problems involving zero, positive and infinite mass cases as well as $G$ need not be even. 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 optimal inequality provides a new proof of the classical logarithmic Sobolev inequality based on a Pohozaev manifold approach. 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 $\mathcal{D}^{1,2}(\mathbb{R}^N)$.