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Multiple solutions for a class of quasilinear problems with double criticality

Part of: Partial differential equations Linear function spaces and their duals Elliptic equations and systems

Published online by Cambridge University Press:  21 October 2022

Karima Ait-Mahiout ,
Claudianor O. Alves  and
Prashanta Garain
Karima Ait-Mahiout
Affiliation:
Laboratoire Théorie du point fixe et Applications, École Normale Supérieure, BP 92, Kouba 16006, Algeria (karima_ait@hotmail.fr)
Claudianor O. Alves
Affiliation:
Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, 58429-970 Campina Grande, PB, Brazil (coalves@mat.ufcg.edu.br)
Prashanta Garain
Affiliation:
Department of Mathematics, Indian Institute of Technology Indore, Khandwa Road, Simrol, Madhya Pradesh 453552, India (pgarain92@gmail.com) Department of Mathematics, Ben-Gurion University of the Negev, P.O.B. 653, Beer Sheva 8410501, Israel, (pgarain92@gmail.com)
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Abstract

We establish multiplicity results for the following class of quasilinear problems P

\begin{equation*} \left\{ \begin{array}{@{}l} -\Delta_{\Phi}u=f(x,u) \quad \mbox{in} \quad \Omega, \\ u=0 \quad \mbox{on} \quad \partial \Omega, \end{array} \right. \end{equation*}
where $\Delta _{\Phi }u=\text {div}(\varphi (x,|\nabla u|)\nabla u)$ for a generalized N-function $\Phi (x,t)=\int _{0}^{|t|}\varphi (x,s)s\,ds$. We consider $\Omega \subset \mathbb {R}^{N}$ to be a smooth bounded domain that contains two disjoint open regions $\Omega _N$ and $\Omega _p$ such that $\overline {\Omega _N}\cap \overline {\Omega _p}=\emptyset$. The main feature of the problem $(P)$ is that the operator $-\Delta _{\Phi }$ behaves like $-\Delta _N$ on $\Omega _N$ and $-\Delta _p$ on $\Omega _p$. We assume the nonlinearity $f:\Omega \times \mathbb {R}\to \mathbb {R}$ of two different types, but both behave like $e^{\alpha |t|^{\frac {N}{N-1}}}$ on $\Omega _N$ and $|t|^{p^{*}-2}t$ on $\Omega _p$ as $|t|$ is large enough, for some $\alpha >0$ and $p^{*}=\frac {Np}{N-p}$ being the critical Sobolev exponent for $1< p< N$. In this context, for one type of nonlinearity $f$, we provide a multiplicity of solutions in a general smooth bounded domain and for another type of nonlinearity $f$, in an annular domain $\Omega$, we establish existence of multiple solutions for the problem $(P)$ that are non-radial and rotationally non-equivalent.

Keywords

variational methodsquasilinear problemsMusielak–Sobolev space

MSC classification

Primary: 35A15: Variational methods 35J62: Quasilinear elliptic equations 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Secondary: 35B33: Critical exponents
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 65 , Issue 4 , November 2022 , pp. 1011 - 1047
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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Footnotes

†

Current address.

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