Singular non-autonomous \((p, q)\)-equations with competing nonlinearities
type: Item
instance of: scholarly article
title: Singular non-autonomous \((p, q)\)-equations with competing nonlinearities
author: Nikolaos S. Papageorgiou
author: Dongdong Qin
author: Vicenţiu D. Rădulescu
published in: Nonlinear Analysis. Real World Applications
publication date: 2024-11-20T00:00:00Z
review text: Let $\Omega \subseteq \mathbb{R}^N$ be a bounded domain with a $C^2$-boundary $\partial \Omega$. In this paper the authors consider a singular Dirichlet problem of the form \N\[ \N\begin{cases}\N-\Delta_{p}^{a_1} u(z)-\Delta_{q}^{a_2} u(z)=\lambda [u(z)^{-\eta} +u(z)^{\tau-1}]+f(z,u(z)) \quad\mbox{in } \Omega, \\\Nu \Big|_{\partial \Omega} =0, \, 0< \eta <1, \, 1<\tau<q<p, \, \lambda>0, \, u > 0.\N\end{cases}\N\]\N\NGiven $1<r <\infty$ and $a \in C^{0,1}(\overline{\Omega})$ with $a(z) \geq c>0$ for all $z \in \overline{\Omega}$, by $\Delta_{r}^a$ we denote the weighted $r$-Laplace differential operator defined by \N\[\N\Delta_{r}^a u= \mbox{div }(a(z)|\nabla u|^{r-2}\nabla u) \mbox{ for all $u \in W_0^{1,r}(\Omega)$.}\N\]\N\NThe perturbation $f(z,x)$ is a Caratheodory function (that is, for all $x \in \mathbb{R}$, $z \to f(z,x)$ is measurable and for a.a. $z \in \Omega$, $x \to f(z,x)$ is continuous) which exhibits $(p-1)$-superlinear growth as $x \to +\infty$, but need not satisfy the usual in such cases Ambrosetti-Rabinowitz condition and may change sign (indefinite perturbation). In the reaction, the authors consider also the effects of two parametric terms: the singular term $u \to \lambda u^{-\eta}$ and the concave term $u \to \lambda u^{\tau-1}$. Hence, the problem in the reaction has the combined effects of singular and concave-convex nonlinearities, however the authors consider also the case when $f(z,x)$ has $(p-1)$-linear growth as $x \to +\infty$. By using appropriate auxiliary problems, hence truncation and comparison arguments, the authors obtain a bifurcation type theorem establishing the existence/non-existence/multiplicity of positive solutions. In addition, the authors study the set of positive solutions $S_\lambda$, and prove the compactness of $S_\lambda$ in $C_0^1(\overline{\Omega})$. Finally, the authors prove suitable continuity properties of the solution multifunction $\lambda \to S_\lambda$.
reviewed by: Calogero Vetro
Mathematics Subject Classification ID: 35J92
Mathematics Subject Classification ID: 35J75
Mathematics Subject Classification ID: 35A01
Mathematics Subject Classification ID: 35A02
zbMATH DE Number: 7947282
zbMATH Keywords: \((p,q)\)-equation
zbMATH Keywords: singular nonlinearities
zbMATH Keywords: concave-convex nonlinearities
zbMATH Keywords: existence
zbMATH Keywords: non-existence
zbMATH Keywords: multiplicity
MaRDI profile type: Publication
cites work: On Lavrentiev's phenomenon
cites work: Q4308736
cites work: Regularity of minimizers of integrals of the calculus of variations with non-standard growth conditions
cites work: Regularity and existence of solutions of elliptic equations with p,q- growth conditions
cites work: On the electrostatic Born-Infeld equation with extended charges
cites work: Double phase transonic flow problems with variable growth: nonlinear patterns and stationary waves
cites work: Solitons in several space dimensions: Derrick's problem and infinitely many solutions
cites work: Combined effects of concave and convex nonlinearities in some elliptic problems
cites work: SOBOLEV VERSUS HÖLDER LOCAL MINIMIZERS AND GLOBAL MULTIPLICITY FOR SOME QUASILINEAR ELLIPTIC EQUATIONS
cites work: \(W^{1,p}\) versus \(C^1\) local minimizers and multiplicity results for quasilinear elliptic equations.
cites work: Positive solutions for nonlinear Robin problems with indefinite potential and competing nonlinearities
cites work: Anisotropic \((p, q)\)-equations with competition phenomena
cites work: On a Dirichlet problem with \((p,q)\)-Laplacian and parametric concave-convex nonlinearity
cites work: Anisotropic equations with indefinite potential and competing nonlinearities
cites work: \((p, q)\)-equations with singular and concave convex nonlinearities
cites work: Singular equations with variable exponents and concave-convex nonlinearities
cites work: Topological properties of the solution sets for parametric nonlinear Dirichlet problems
cites work: Positive and nodal solutions for parametric superlinear weighted \(( p , q )\)-equations
cites work: Continuity results for parametric nonlinear singular Dirichlet problems
cites work: Nonlinear Analysis - Theory and Methods
cites work: Q6107004
cites work: The natural generalizationj of the natural conditions of ladyzhenskaya and uralľtseva for elliptic equations
cites work: The maximum principle
cites work: Exercises in analysis. II: Nonlinear analysis
cites work: Research Topics in Analysis, Volume I
cites work: Minimax theorems
cites work: \(L^\infty\)-bounds for general singular elliptic equations with convection term
cites work: Q3803507
cites work: Coercive and noncoercive nonlinear Neumann problems with indefinite potential
cites work: Positive solutions for the Robin \(p\)-Laplacian problem with competing nonlinearities
cites work: A bifurcation-type theorem for singular nonlinear elliptic equations
cites work: Uniqueness of positive solutions for quasilinear elliptic equations when a parameter is large
cites work: Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints
cites work: Q4064235
cites work: Global multiplicity for the positive solutions of parametric singular \((p, q)\)-equations with indefinite perturbations
cites work: Sobolev and Hölder regularity results for some singular nonhomogeneous quasilinear problems
cites work: Anisotropic singular double Phase Dirichlet problems
cites work: Elliptic partial differential equations of second order
cites work: Nonlinear nonhomogeneous singular problems
cites work: Multiple constant sign and nodal solutions for nonlinear elliptic equations with the \(p\)-Laplacian
cites work: Positive solutions for nonlinear singular problems with sign-changing nonlinearities
cites work: \(W^{1,p}_0\) versus \(C^1\) local minimizers for a singular and critical functional
DOI: 10.1016/J.NONRWA.2024.104225
description: scientific article; zbMATH DE number 7947282