Multiplicity and concentration results for a ▫$(p, q)$▫-Laplacian problem in ▫${\mathbb{R}}^N$▫
Ambrosio, Vincenzo (Author), Repovš, Dušan (Author)

Abstract
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 (p, q)-Laplacian problem, positive solutions, variational methods, Ljusternik-Schnirelmann theory Article (dk_c) 1.01 - Original Scientific Article PEF - Faculty of EducationFMF - Faculty of Mathematics and Physics 2021 art. 33 (33 str.) Vol. 72, iss. 1 517.956 0044-2275 10.1007/s00033-020-01466-7 47945731 213 83 (0 votes) Voting is allowed only to logged in users. AddThis uses cookies that require your consent. Edit consent...

## Record is a part of a journal

Title: Zeitschrift für angewandte Mathematik und Physik Z. angew. Math. Phys. Birkhaeuser Verlag 0044-2275 26662656

## Document is financed by a project

Funder: ARRS - Agencija za raziskovalno dejavnost Republike Slovenije (ARRS) P1-0292 P1-0292 Topologija, geometrija in nelinearna analiza

## Licences

License: CC BY 4.0, Creative Commons Attribution 4.0 International http://creativecommons.org/licenses/by/4.0/ 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. 25.01.2021