The paper concerns with positive solutions of problems of the type $-Delta u+a(x), u=u^{p-1}+arepsilon u^{2^*-1}$ in $OmegasubseteqR^N$, $Nge 3$, $2^*={2Nover N-2}$, $2<2^*$. Here $Omega$ can be an exterior domain, i.e. $R^NsetminusOmega$ is bounded, or the whole of $R^N$. The potential $ain L^{N/2}_{loc}(R^N)$ is assumed to be strictly positive and such that there exists $lim_{|x| oinfty}a(x):=a_infty>0$. First, some existence results of ground state solutions are proved. Then the case $a(x)ge a_infty$ is considered, with $a(x) otequiv a_infty$ or $Omega eqR^N$. In such a case, no ground state solution exists and the existence of a bound state solution is proved, for small $e$.

Lancelotti, S., Molle, R. (2020). Positive solutions for autonomous and non-autonomous nonlinear critical elliptic problems in unbounded domains. NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS, 27(1) [10.1007/s00030-019-0611-5].

Positive solutions for autonomous and non-autonomous nonlinear critical elliptic problems in unbounded domains

Molle R.
2020-01-01

Abstract

The paper concerns with positive solutions of problems of the type $-Delta u+a(x), u=u^{p-1}+arepsilon u^{2^*-1}$ in $OmegasubseteqR^N$, $Nge 3$, $2^*={2Nover N-2}$, $2<2^*$. Here $Omega$ can be an exterior domain, i.e. $R^NsetminusOmega$ is bounded, or the whole of $R^N$. The potential $ain L^{N/2}_{loc}(R^N)$ is assumed to be strictly positive and such that there exists $lim_{|x| oinfty}a(x):=a_infty>0$. First, some existence results of ground state solutions are proved. Then the case $a(x)ge a_infty$ is considered, with $a(x) otequiv a_infty$ or $Omega eqR^N$. In such a case, no ground state solution exists and the existence of a bound state solution is proved, for small $e$.
2020
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/05 - ANALISI MATEMATICA
English
Con Impact Factor ISI
Schrödinger equations; exterior domains; critical nonlinearity
The authors have been supported by the ``Gruppo Nazionale per l'Analisi Matematica, la Probabilit`a e le loro Applicazioni (GNAMPA)'' of the {em Istituto Nazionale di Alta Matematica (INdAM)} - Project: Sistemi differenziali ellittici nonlineari derivanti dallo studio di fenomeni elettromagnetici. oindent The first author acknowledges also the MIUR Excellence Department Project awarded to the Department of Mathematical Sciences, Politecnico of Turin, CUP E11G18000350001. oindent The second author acknowledges also the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.
Lancelotti, S., Molle, R. (2020). Positive solutions for autonomous and non-autonomous nonlinear critical elliptic problems in unbounded domains. NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS, 27(1) [10.1007/s00030-019-0611-5].
Lancelotti, S; Molle, R
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/262579
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