Multiplicity and concentration results for a $(p, q)$-Laplacian problem in ${\mathbb{R}}^N$

Ambrosio, Vincenzo (author), Repovš, Dušan (author)

In this paper, we study the multiplicity and concentration of positive solutions for the following $(p, q)$-Laplacian problem: $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{p} u -\Delta _{q} u +V(\varepsilon x) \left( |u|^{p-2}u + |u|^{q-2}u\right) = f(u) &{} \text{ in } {\mathbb{R}}^{N}, \\ u\in W^{1, p}({\mathbb{R}}^{N})\cap W^{1, q}({\mathbb{R}}^{N}), \quad u>0 \text{ in } {\mathbb{R}}^{N}, \end{array} \right. \end{aligned}$$ where $\varepsilon >0$ is a small parameter, $1 < p < q < N$, $\Delta _{r}u={{\,\mathrm{div}\,}}(|\nabla u|^{r-2}\nabla u)$, with $r\in \{p, q\}$, is the $r$-Laplacian operator, $V:{\mathbb{R}}^{N}\rightarrow {\mathbb{R}}$ is a continuous function satisfying the global Rabinowitz condition, and $f:{\mathbb{R}}\rightarrow {\mathbb{R}}$ is a continuous function with subcritical growth. Using suitable variational arguments and Ljusternik-Schnirelmann category theory, we investigate the relation between the number of positive solutions and the topology of the set where $V$ attains its minimum for small $\varepsilon$.

• Language: English
• Keywords: (p, q)-Laplacian problem, positive solutions, variational methods, Ljusternik-Schnirelmann theory
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: PEF - Faculty of Education; FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Author Accepted Manuscript
• Year: 2021
• Number of pages: art. 33 (33 str.)
• Numbering: Vol. 72, iss. 1
• PID: 20.500.12556/RUL-124471
• UDC: 517.956
• ISSN on article: 0044-2275
• DOI: 10.1007/s00033-020-01466-7
• COBISS.SI-ID: 47945731
• Publication date in RUL: 25.01.2021

AMBROSIO, Vincenzo and REPOVŠ, Dušan, 2021, Multiplicity and concentration results for a ▫$(p, q)$▫-Laplacian problem in ▫${\mathbb{R}}^N$▫. Zeitschrift für angewandte Mathematik und Physik [online]. 2021. Vol. 72, no. 1. [Accessed 14 April 2025]. DOI 10.1007/s00033-020-01466-7. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=124471

Record is a part of a journal

• Title: Zeitschrift für angewandte Mathematik und Physik
• Shortened title: Z. angew. Math. Phys.
• Publisher: Birkhaeuser Verlag
• ISSN: 0044-2275
• COBISS.SI-ID: 26662656

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.
• Licensing start date: 25.01.2021

Projects

• Funder: ARRS - Slovenian Research Agency
• Funding programme: P1-0292
• Project number: P1-0292
• Name: Topologija, geometrija in nelinearna analiza