[{"type":"report","title":"General class of optimal Sobolev inequalities and nonlinear scalar field equations","issued":{"date-parts":[["2018"]]},"DOI":"10.5445\/IR\/1000089034","author":[{"family":"Mederski","given":"Jaros\u0142aw"}],"publisher":"Karlsruher Institut f\u00fcr Technologie (KIT)","ISSN":"2365-662X","collection-title":"CRC 1173","abstract":"We find a class of optimal Sobolev inequalities\r\n$$\\left(\\int_{\\mathbb{R}^N}|\\nabla u|^2\\,dx\\right)^{\\frac{N}{N-2}}\\geq C_{N,G}\\int_{\\mathbb{R}^N}G(u)\\,dx,\\quad u\\in\\mathcal{D}^{1,2}(\\mathbb{R}^N), N\\geq3$$\r\nwhere 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 $$-\\Delta u=g(u)\\quad\\text{in }\\mathbb{R}^N,\\quad\\text{with }g=G'.$$\r\nIn particular, if $G(u)=u^2\\DeclareMathOperator{\\log}{log}\\log|u|$, then the sharp constant is $C_{N,G}:=2*\\left(\\frac{N}{2}\\right) e^{\\frac{2(N-1)}{N-2}}(\\pi)^{\\frac{N}{N-2}}$ and $u_{\\lambda}(x) = e^{\\frac{N-1}{N}-\\frac{\\lambda\u00b2}{2}|x|^2}$ with $\u03bb>0$ constitutes the whole family of minimizers up to translations. The above optimal inequality provides a simple proof of the classical logarithmic Sobolev inequality.\r\nMoreover, if $N\\geq4$, then there is at least one nonradial solution and if, in addition, $N\\neq5$, then there are infinitely many nonradial solutions of the nonlinear scalar field equation. The energy functional associated with the problem may be infinite on $D^{1,2}(\\mathbb{R}^N)$ and is not Fr\u00e9chet differentiable in its domain. We present a variational approach to this problem based on a new variant of Lions\u2019 lemma in $D^{1,2}(\\mathbb{R}^N)$.","keyword":"nonlinear scalar field equations, logarithmic Sobolev inequality, cubic-quintic effect, critical point theory, nonradial solutions, concentration compactness, Lions\u2019 lemma, Pohozaev manifold, zero mass case, infinite mas case","number-of-pages":28,"kit-publication-id":"1000089034"}]