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Multiplicity and concentration results for a ▫$(p, q)$▫-Laplacian problem in ▫${\mathbb{R}}^N$▫
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Ambrosio, Vincenzo
(
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),
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Repovš, Dušan
(
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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
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
Views:
3738
Downloads:
214
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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
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