Abstract multiplicity results for \((p, q)\)-Laplace equations with two parameters
type: Item
instance of: scholarly article
title: Abstract multiplicity results for \((p, q)\)-Laplace equations with two parameters
author: Vladimir Bobkov
author: Mieko Tanaka
publication date: 2024-11-29T00:00:00Z
review text: The paper considers the Dirichlet problem for the equation \[-\Delta_p u -\Delta_q u = \alpha |u|^{ p-2} u + \beta |u| ^{q-2} u\] in a bounded smooth domain, with \(1 < q < p\) and \(\alpha,\beta\in\mathbb{R}\). The aim is to obtain multiplicity of solutions (either positive or sign changing) for suitable combination of the parameters \(\alpha,\beta\).\N\NSolutions, either with positive or with negative energy, are obtained via variational methods and in particular using a Nehari manifold analysis and Ljusternik-Schnirelmann theory.\N\NThe conditions on \(\alpha,\beta\) are related to the variational eigenvalues of \(-\Delta_p\) and \(-\Delta_q\), and to the following critical parameters \[ \beta_*(\alpha)=\inf \left\{\frac{\|\nabla u\|_q^q}{\|u\|_q^q}: u \in W_0^{1, p} \backslash\{0\} \text { and } \|\nabla u\|_p^p\leq \alpha\|u) \|_p^p\right\}. \]
reviewed by: Eugenio Massa
Mathematics Subject Classification ID: 35J92
Mathematics Subject Classification ID: 35B30
Mathematics Subject Classification ID: 35A01
Mathematics Subject Classification ID: 35B38
zbMATH DE Number: 7951505
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: \((p, q)\)-Laplacian
zbMATH Keywords: variational eigenvalues
zbMATH Keywords: multiplicity
zbMATH Keywords: symmetric mountain pass theorem
zbMATH Keywords: Nehari manifold
MaRDI profile type: Publication
cites work: On semilinear elliptic equations with indefinite nonlinearities
cites work: The Asymptotic Behavior of the Solutions of Degenerate Parabolic Equations
cites work: Variational methods for indefinite superlinear homogeneous elliptic problems
cites work: Improved Friedrichs inequality for a subhomogeneous embedding
cites work: On positive solutions for \((p,q)\)-Laplace equations with two parameters
cites work: On sign-changing solutions for \((p,q)\)-Laplace equations with two parameters
cites work: Remarks on minimizers for \((p,q)\)-Laplace equations with two parameters
cites work: Asymptotic behaviour of nonlinear eigenvalue problems involving $p$-Laplacian-type operators
cites work: Q5636090
cites work: Multiple solutions for asymptotically q‐linear (p, q)‐Laplacian problems
cites work: Eigenvalues for double phase variational integrals
cites work: On the dependence on \(p\) of the variational eigenvalues of the \(p\)-Laplace operator
cites work: On the existence of positive solutions and solutions with compact support for a spectral nonlinear elliptic problem with strong absorption
cites work: Variational and topological methods for Dirichlet problems with \(p\)-Laplacian
cites work: Positive solutions for the p-Laplacian: application of the fibrering method
cites work: An improved Poincaré inequality and the \(p\)-Laplacian at resonance for \(p>2\)
cites work: LJUSTERNIK‐SCHNIRELMANN Theorem and Nonlinear Eigenvalue Problems
cites work: On the stability of Sobolev spaces with zero boundary values
cites work: On positive solutions of indefinite elliptic equations
cites work: On branches of positive solutions for 𝑝-Laplacian problems at the extreme value of the Nehari manifold method
cites work: Q4963470
cites work: Uniqueness and positivity issues in a quasilinear indefinite problem
cites work: Boundary regularity for solutions of degenerate elliptic equations
cites work: The natural generalizationj of the natural conditions of ladyzhenskaya and uralľtseva for elliptic equations
cites work: On nonlinear Rayleigh quotients
cites work: Some recent results on the Dirichlet problem for \((p,q)\)-Laplace equations
cites work: CONTINUITY OF THE VARIATIONAL EIGENVALUES OF THE p-LAPLACIAN WITH RESPECT TO p
cites work: Q3560017
cites work: Local minimizers for a class of functionals over the Nehari set
cites work: Q3749777
cites work: Q5595759
cites work: Variational Methods
cites work: Ljusternik-Schnirelmann theory on \(C^ 1\)-manifolds
cites work: Generalized eigenvalue problems for \((p,q)\)-Laplacian with indefinite weight
cites work: A property of Sobolev spaces and existence in optimal design
cites work: Ljusternik-Schnirelmann theorem for the generalized Laplacian
cites work: Q3706043
cites work: Nonlinear eigenvalue problems and bifurcation for quasi-linear elliptic operators
published in: Rendiconti del Circolo Matematico di Palermo
DOI: 10.1007/S12215-024-01067-7
description: scientific article; zbMATH DE number 7951505