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General class of optimal Sobolev inequalities and nonlinear scalar field equations

Mederski, Jarosław


We find a class of optimal Sobolev inequalities
$$\left(\int_{\mathbb{R}^N}|\nabla u|^2\,dx\right)^{\frac{N}{N-2}}\geq C_{N,G}\int_{\mathbb{R}^N}G(u)\,dx,\quad u\in\mathcal{D}^{1,2}(\mathbb{R}^N), N\geq3$$
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 $$-\Delta u=g(u)\quad\text{in }\mathbb{R}^N,\quad\text{with }g=G'.$$
In 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 $u_{\lambda}(x) = e^{\frac{N-1}{N}-\frac{\lambda²}{2}|x|^2}$ with $λ>0$ 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 $N\geq4$, 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. ... mehr

Volltext §
DOI: 10.5445/IR/1000089034
Veröffentlicht am 03.01.2019
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Analysis (IANA)
Sonderforschungsbereich 1173 (SFB 1173)
Publikationstyp Forschungsbericht/Preprint
Publikationsjahr 2018
Sprache Englisch
Identifikator ISSN: 2365-662X
KITopen-ID: 1000089034
Verlag Karlsruher Institut für Technologie (KIT)
Umfang 28 S.
Serie CRC 1173 ; 2018/55
Projektinformation SFB 1173/1 (DFG, DFG KOORD, SFB 1173/1 2015)
Schlagwörter nonlinear scalar field equations, logarithmic Sobolev inequality, cubic-quintic effect, critical point theory, nonradial solutions, concentration compactness, Lions’ lemma, Pohozaev manifold, zero mass case, infinite mas case
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