Multiplicity and concentration results for a (pq)-Laplacian problem in \({\mathbb {R}}^{N}\)

Abstract

In this paper, we study the multiplicity and concentration of positive solutions for the following (pq)-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 \(\varepsilon \).

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Acknowledgements

The authors are very grateful to the anonymous referee for his/her careful reading of the manuscript and valuable suggestions that improved the presentation of the paper. The first author was partly supported by the GNAMPA Project 2020 entitled: Studio Di Problemi Frazionari Nonlocali Tramite Tecniche Variazionali. The second author was partly supported by Slovenian research agency Grants P1-0292, N1-0114, N1-0083, N1-0064 and J1-8131.

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Correspondence to Vincenzo Ambrosio.

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Ambrosio, V., Repovš, D. Multiplicity and concentration results for a (pq)-Laplacian problem in \({\mathbb {R}}^{N}\). Z. Angew. Math. Phys. 72, 33 (2021). https://doi.org/10.1007/s00033-020-01466-7

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Keywords

  • (p
  •  q)-Laplacian problem
  • Positive solutions
  • Variational methods
  • Ljusternik–Schnirelmann theory

Mathematics Subject Classification

  • 35A15
  • 35B09
  • 35J62
  • 58E05