General class of optimal Sobolev inequalities and nonlinear scalar field equations

Abstract

We find a class of optimal Sobolev inequalities(∫RN|∇u|2dx)NN−2≥CN,G∫RNG(u)dx,u∈D1,2(RN),N≥3, where the nonlinear function G:R→R of class C1 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−Δu=g(u)in RN,with g=G′. In particular, if G(u)=u2log⁡|u|, then the sharp constant is CN,G:=2⁎(N2)2N−2e2(N−1)N−2(π)NN−2 and uλ(x)=eN−12−λ22|x|2 with λ>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≥4, then there is at least one nonradial solution and if, in addition, N≠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 D1,2(RN) and is not Fréchet differentiable in its domain. We present a variational approach to this problem based on a new variant of Lions' lemma in D1,2(RN).