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

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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
Keywords:(p, q)-Laplacian problem, positive solutions, variational methods, Ljusternik-Schnirelmann theory
Work type:Article (dk_c)
Tipology:1.01 - Original Scientific Article
Organization:PEF - Faculty of Education
FMF - Faculty of Mathematics and Physics
Year:2021
Number of pages:art. 33 (33 str.)
Numbering:Vol. 72, iss. 1
UDC:517.956
ISSN on article:0044-2275
DOI:10.1007/s00033-020-01466-7 This link opens in a new window
COBISS.SI-ID:47945731 This link opens in a new window
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Downloads:83
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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 This link opens in a new window

Document is financed by a project

Funder:ARRS - Agencija za raziskovalno dejavnost Republike Slovenije (ARRS)
Funding Programme:P1-0292
Project no.:P1-0292
Name:Topologija, geometrija in nelinearna analiza

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

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