Multiple positive solutions for a p-Laplace Benci–Cerami type problem (1<2), via Morse theory
| Autor: | Giuseppina Vannella |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2023 |
| Předmět: |
Pure mathematics
Laplace transform Computer Science::Information Retrieval Applied Mathematics General Mathematics Astrophysics::Instrumentation and Methods for Astrophysics 58E05 35J60 35J92 35B20 Boundary (topology) Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) Type (model theory) Domain (mathematical analysis) TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES Mathematics - Analysis of PDEs Bounded function ComputingMethodologies_DOCUMENTANDTEXTPROCESSING Computer Science::General Literature p-Laplace equations Morse theory perturbation results critical groups ComputingMilieux_MISCELLANEOUS Mathematics |
| Popis: | Let us consider the quasilinear problem \[ (P_\varepsilon) \ \ \left\{ \begin{array}{ll} - \varepsilon^p \Delta _{p}u + u^{p-1} = f(u) & \hbox{in} \ \Omega \newline u>0 & \hbox{in} \ \Omega \newline u=0 & \hbox{on} \ \partial \Omega \end{array} \right. \] where $\Omega$ is a bounded domain in $\mathbb{R}^N$ with smooth boundary, $N\geq 2$, $1< p < 2$, $\varepsilon >0$ is a parameter and $f: \mathbb{R} \to \mathbb{R}$ is a continuous function with $f(0)=0$, having a subcritical growth. We prove that there exists $\varepsilon^* >0$ such that, for every $\varepsilon \in (0, \varepsilon^*)$, $(P_\varepsilon)$ has at least $2{\mathcal P}_1(\Omega)-1$ solutions, possibly counted with their multiplicities, where ${\mathcal P}_t(\Omega)$ is the Poincar\'e polynomial of $\Omega$. Using Morse techniques, we furnish an interpretation of the multiplicity of a solution, in terms of positive distinct solutions of a quasilinear equation on $\Omega$, approximating $(P_\varepsilon)$. Comment: to be published in "Communications in Contemporary Mathematics" |
| Databáze: | OpenAIRE |
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