Existence and multiplicity of solutions for a class of elliptic problem with critical growth
type: Item
instance of: scholarly article
title: Existence and multiplicity of solutions for a class of elliptic problem with critical growth
zbMATH Open document ID: 1421.35142
author: Luciano M. Barros
author: Claudianor O. Alves
published in: Monatshefte für Mathematik
publication date: 2019-03-01T00:00:00Z
review text: The authors study a class of elliptic problems of the type \begin{align*} -\Delta u +\lambda V(x)u=\mu u^{q-1}+u^{2^*-1} \quad\text{in }\mathbb{R}^N\tag{P} \end{align*} where $\lambda, \mu >0$, $q\in (2,2^*)$ with $N\geq 3$ and $V: \mathbb{R}^N\to \mathbb{R}$ is a continuous function that satisfies \begin{itemize} \item[(V1)] $V(x) \geq 0$ for all $x\in \mathbb{R}^N$. \item[(V2)] There exists $M_0>0$ such that the set $\mathcal{L}=\{x\in \mathbb{R}^N\, : \, V(x) \leq M_0\}$ is non-empty and $|\mathcal{L}|<\infty,$ where $|A|$ denotes the Lebesgue measure of $A$ on $\mathbb{R}^N$. \item[(V3)] $\Omega:=\text{int\,}\left(V^{-1}(\{0\}) \right)$ is a non-empty bounded open set with smooth boundary $\partial\Omega$. \end{itemize} \par The main result in this paper, which proof is based on variational methods, is the following. \par Theorem 1.1. Assume that (V1)--(V3) hold. If \begin{align*} N\geq 4 \text{ and }2<q<2^*\quad\text{or}\quad N=3 \text{ and }4<q<6, \end{align*} then there are $\lambda^*, \mu^*>0$ such that problem (P) has at least $\text{cat\,}(\Omega)$ positive solutions for $\lambda\geq \lambda^*$ and $\mu\leq \mu^*$.
reviewed by: Patrick Winkert
Mathematics Subject Classification ID: 35J91
Mathematics Subject Classification ID: 35B09
Mathematics Subject Classification ID: 35B33
Mathematics Subject Classification ID: 35A15
zbMATH DE Number: 7031532
zbMATH Keywords: semilinear equation with Laplacian
zbMATH Keywords: critical exponents
zbMATH Keywords: positive solutions
zbMATH Keywords: variational methods
MaRDI profile type: Publication
full work available at URL: s00605-017-1117-z
OpenAlex ID: W2765555866
cites work: Multiplicity of positive solutions to a \(p\)-Laplacian equation involving critical nonlinearity.
cites work: Multiplicity of positive solutions for a class of problems with critical growth in ℝN
cites work: Multi-bump solutions for Choquard equation with deepening potential well
cites work: Multiplicity of positive solutions for a class of problems with exponential critical growth in \(\mathbb R^2\)
cites work: Dual variational methods in critical point theory and applications
cites work: NONLINEAR SCHRÖDINGER EQUATIONS WITH STEEP POTENTIAL WELL
cites work: Multibump solutions of nonlinear Schrödinger equations with steep potential well and indefinite potential
cites work: Existence and multiplicity results for some superlinear elliptic problems on RN
cites work: Multiple positive solutions for a nonlinear Schrödinger equation
cites work: Positive solutions of a Schrödinger equation with critical nonlinearity
cites work: Multiplicity of positive solutions of a nonlinear Schrödinger equation
cites work: Q5187597
cites work: Existence of multi-bumb solutions for nonlinear schrödinger equations via variational method
cites work: Multi-bump solutions for Schrödinger equation involving critical growth and potential wells
cites work: On a class of semilinear elliptic problems in ℝN with critical growth
cites work: Existence and symmetry of multi-bump solutions for nonlinear Schrödinger equations
cites work: Minimax theorems
DOI: 10.1007/S00605-017-1117-Z
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description: scientific article; zbMATH DE number 7031532