The existence of a nontrivial solution to the \(p{\&}q\)-Laplacian problem with nonlinearity asymptotic to \(u^{p - 1}\) at infinity in \(\mathbb R^N\)
type: Item
instance of: scholarly article
title: The existence of a nontrivial solution to the \(p{\&}q\)-Laplacian problem with nonlinearity asymptotic to \(u^{p - 1}\) at infinity in \(\mathbb R^N\)
zbMATH Open document ID: 1148.35024
author: Gongbao Li
author: Cheng-Jun He
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2008-03-06T00:00:00Z
review text: Let \(m,n>0\), \(N\geq3\) and \(1<q<p<N\). Let \(\Delta_p u=\text{div}(|\nabla u|^{s-2}\nabla u)\) denote the \(p\)-Laplacian of \(u\). The nonlinear elliptic problem \[ -\Delta_p u+m| u|^{p-2}u-\Delta_q u+n| u|^{q-2}u=f(x,u),\quad x\in\mathbb R^N, \] is considered, where \(f:\mathbb R^N\times\mathbb R\to\mathbb R\) does not satisfy the usual requirement, i.e., the Ambrosetti-Rabinowitz condition \(0\leq\int_0^u f(x,t)\,dt\leq (p+\theta)^{-1}f(x,u)u\). Instead, it is assumed that \(f\) is a non-negative function satisfying the Carathéodory conditions, \(f(x,t)=0\) if \(t<0\), \(f(x,t)t^{1-p}\) tends to zero as \(t\to0_+\) and to a positive constant as \(t\to\infty\), both uniformly in \(x\in\mathbb R^N\), \(f(x,t)t^{1-p}\) is nondecreasing in \(t\) for every \(x\in\mathbb R^N\) and satisfies some additional conditions. The authors mainly follow the approach of \textit{G. Li} and \textit{H.-S. Zhou} [Proc. R. Soc. Edinb., Sect. A, Math. 130, No. 1, 81--105 (2000; Zbl 0942.35075)] to prove existence of a nontrivial solution \(u\in W^{1,p}(\mathbb R^N)\cap W^{1,q}(\mathbb R^N)\).
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35D05
zbMATH DE Number: 5243587
zbMATH Keywords: \(p\)-\(q\)-Laplacian
zbMATH Keywords: nontrivial solution
zbMATH Keywords: existence
reviewed by: Jiří Rákosník
MaRDI profile type: Publication
full work available at URL: j.na.2006.12.008
OpenAlex ID: W1488099784
cites work: Dual variational methods in critical point theory and applications
cites work: An eigenvalue problem for a quasilinear elliptic field equation
cites work: Positive solutions of nonlinear elliptic equations involving critical sobolev exponents
cites work: Q3999077
cites work: On the stationary solutions of generalized reaction diffusion equations with \(p\)\& \(q\)-Laplacian
cites work: Nontrivial solutions for perturbations of the \(p\)-Laplacian on unbounded domains
cites work: On the existence of positive entire solutions of a semilinear elliptic equation
cites work: On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on ℝN
cites work: AN ASYMPTOTICALLY PERIODIC SCHRÖDINGER EQUATION WITH INDEFINITE LINEAR PART
cites work: The existence of a positive solution to asymptotically linear scalar field equations
cites work: Q3832935
cites work: Q5749694
cites work: The concentration-compactness principle in the calculus of variations. The locally compact case. II
cites work: The concentration-compactness principle in the calculus of variations. The locally compact case. I
cites work: Existence of solitary waves in higher dimensions
cites work: A Variational Problem Related to Self-trapping of an Electromagnetic Field
cites work: Applying the mountain pass theorem to an asymptotically linear elliptic equation on RN
cites work: A strong maximum principle for some quasilinear elliptic equations
cites work: Q3828389
DOI: 10.1016/J.NA.2006.12.008
Recommended article: L&infin;-Bounds of Solutions for Strongly Nonlinear Elliptic Problems with Two Lower Order Terms
Recommended article: Existence of a nontrivial solution for the \((p, q)\)-Laplacian in \(\mathbb{R}^N\) without the Ambrosetti-Rabinowitz condition
Recommended article: Q3485282
Recommended article: Existence of a nontrivial solution for a \((p,q)\)-Laplacian equation with \(p\)-critical exponent in \(\mathbb{R}^{N}\)
Recommended article: Sum of weighted Lebesgue spaces and nonlinear elliptic equations
Recommended article: Increase of powers in the lower order term: a come back when the source term has a poor summability
Recommended article: Variational approach to MEMS model with fringing field
Recommended article: On a class of quasilinear eigenvalue problems on unbounded domains
Recommended article: Existence results for nonlinear elliptic equations with degenerate coercivity
Recommended article: Q4033556
description: scientific article