Existence results for a class of local and nonlocal nonlinear elliptic problems

Said Ait Temghart; Chakir Allalou; Adil Abbassi

doi:10.56754/0719-0646.2501.001

DOI:

https://doi.org/10.56754/0719-0646.2501.001

Abstract

In this paper, we study the \(p\)-Laplacian problems in the case where \(p\) depends on the solution itself. We consider two situations, when \(p\) is a local and nonlocal quantity. By using a singular perturbation technique, we prove the existence of weak solutions for the problem associated to the following equation

\[\begin{cases}-\mathrm{d}\mathrm{i}\mathrm{v}(|\nabla u|^{p(u)-2}\nabla u)+|u|^{p(u)-2}u=f&\mbox{in}\; \Omega\\u=0& \mbox{on}\; \partial\Omega,\end{cases}\]

and also for its nonlocal version. The main goal of this paper is to extend the results established by M. Chipot and H. B. de Oliveira (Math. Ann., 2019, 375, 283-306).

Keywords

p(u)-Laplacian , elliptic problems , variable nonlinearity , generalised Sobolev spaces

Mathematics Subject Classification:

35J60 , 35J05 , 35D30

Said Ait Temghart Laboratory LMACS, FST of Beni Mellal, Sultan Moulay Slimane University, Morocco. https://orcid.org/0000-0002-7962-2296
Chakir Allalou Laboratory LMACS, FST of Beni Mellal, Sultan Moulay Slimane University, Morocco. https://orcid.org/0000-0002-4885-9397
Adil Abbassi Laboratory LMACS, FST of Beni Mellal, Sultan Moulay Slimane University, Morocco. https://orcid.org/0000-0003-0284-8390

Pages: 1–21
Date Published: 2023-04-24
Vol. 25 No. 1 (2023)

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