Multiple solutions with sign information for a class of parametric superlinear \((p, 2)\)-equations
type: Item
instance of: scholarly article
title: Multiple solutions with sign information for a class of parametric superlinear \((p, 2)\)-equations
zbMATH Open document ID: 1479.35294
author: Leszek Gasiński
author: Nikolaos S. Papageorgiou
published in: Applied Mathematics and Optimization
publication date: 2021-07-15T00:00:00Z
review text: The authors consider the following parametric $(p, 2)$-equation (two-phase problem): \[ \begin{cases} -\Delta_p u(z)-\Delta u(z)=\lambda |u(z)|^{r-2}u(z)+f(z, u(z))\quad\text{in }\Omega,\\ u|_{\partial\Omega}=0,\ 2<p<r<p^*,\ \lambda>0, \end{cases}\tag{1} \] where $\Omega$ is a bounded domain of $\mathbb{R}^N$ with a $C^2$-boundary $\partial\Omega$ and $p^*$ is the critical Sobolev exponent corresponding to $p$. Using variational tools from the critical point theory together with suitable truncation and comparison techniques and critical groups (Morse theory), the authors show that for all $\lambda > 0$ big, problem (1) has at least three nontrivial smooth solutions all with sign information. If they strengthen the regularity of $f(z,\cdot)$, they prove the existence of a second nodal solution, for a total of four nontrivial smooth solutions, all with sign information.
reviewed by: Said El Manouni
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 7371818
zbMATH Keywords: two-phase problem
zbMATH Keywords: constant sign solutions
zbMATH Keywords: extremal solutions
zbMATH Keywords: nodal solutions
zbMATH Keywords: nonlinear regularity
zbMATH Keywords: comparison principle
zbMATH Keywords: asymptotic behaviour
zbMATH Keywords: critical groups
MaRDI profile type: Publication
full work available at URL: s00245-019-09595-w
OpenAlex ID: W2962999884
Wikidata QID: Q112251707
cites work: Nonlinear nonhomogeneous Robin problems with dependence on the gradient
cites work: Q5376881
cites work: Nonlinear nonhomogeneous Robin problems with convection
cites work: On the stationary solutions of generalized reaction diffusion equations with \(p\)\& \(q\)-Laplacian
cites work: Q5350064
cites work: Multiple constant sign and nodal solutions for nonlinear elliptic equations with the \(p\)-Laplacian
cites work: Q5312447
cites work: Multiple solutions for nonlinear coercive problems with a nonhomogeneous differential operator and a nonsmooth potential
cites work: A pair of positive solutions for \((p, q)\)-equations with combined nonlinearities
cites work: Dirichlet (\(p,q\))-equations at resonance
cites work: Exercises in analysis. II: Nonlinear analysis
cites work: Asymmetric \(8(p,2)\)-equations with double resonance
cites work: Q5000544
cites work: Positive solutions for the Robin \(p\)-Laplacian problem with competing nonlinearities
cites work: Nonlinear Dirichlet problems with sign changing drift coefficient
cites work: Multiple solutions for \((p,2)\)-equations with resonance and concave terms
cites work: Q4367585
cites work: Linear and quasilinear elliptic equations
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: Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems
cites work: On Parametric Evolution Inclusions of the Subdifferential Type with Applications to Optimal Control Problems
cites work: Optimal control and admissible relaxation of uncertain nonlinear elliptic systems
cites work: Qualitative phenomena for some classes of quasilinear elliptic equations with multiple resonance
cites work: Bifurcation of positive solutions for nonlinear nonhomogeneous Robin and Neumann problems with competing nonlinearities
cites work: Nonlinear nonhomogeneous Robin problems with superlinear reaction term
cites work: On a class of parametric \((p,2)\)-equations
cites work: Nonlinear Analysis - Theory and Methods
cites work: Nonlinear Robin problems with general potential and crossing reaction
cites work: Constant sign and nodal solutions for parametric \((p, 2)\)-equations
cites work: The maximum principle
cites work: Optimal control in coefficients for weak variational problems in Hilbert space
cites work: On continuity of minimizers for certain quadratic growth functionals
cites work: Boundary regularity of minimizers of \(p(x)\)-energy functionals
cites work: Nonlinear elliptic problem of 2-\(q\)-Laplacian type with asymmetric nonlinearities
cites work: AVERAGING OF FUNCTIONALS OF THE CALCULUS OF VARIATIONS AND ELASTICITY THEORY
cites work: On variational problems and nonlinear elliptic equations with nonstandard growth conditions
DOI: 10.1007/S00245-019-09595-W
description: scientific article; zbMATH DE number 7371818