Special Issue in honor of Alan C. Lazer. Electron. J. Diff. Eqns., Special Issue 01 (2021), pp. 327-344.

Generalized quasilinear equations with critical growth and nonlinear boundary conditions

Liliane de Almeida Maia, Jose Carlos Oliveira Junior, Ricardo Ruviaro

Abstract:
We study the quasilinear problem

where $\Omega \subset \mathbb{R}^3$ is a bounded domain with regular boundary $\partial \Omega$, $\lambda,\mu>0$, $1<q<4$, $2\cdot2^{\ast}=12$, $\frac{\partial }{\partial\eta}$ is the outer normal derivative and g has a subcritical growth in the sense of the trace Sobolev embedding. We prove a regularity result for all weak solutions for a modified, and introducing a new type of constraint, we obtain a multiplicity of solutions, including the existence of a ground state.

Published June 27, 2022.
Math Subject Classifications: 35J25, 35J62, 35B33.
Key Words: Quasilinear equations; variational methods; concave nonlinearities critical exponent; ground state solution.
DOI: https://doi.org/10.58997/ejde.sp.01.m3

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Liliane de A. Maia
Universidade de Brasília
Departamento de Matemática, 70.910-900
Brasília, DF, Brazil
email: lilimaia@unb.br
José Carlos Oliveira Junior
Universidade Federal do Tocantins
Departamento de Matemática, 77.824-838
Araguaína, TO, Brazil
email: jc.oliveira@uft.edu.br
Ricardo Ruviaro
Universidade de Brasília
Departamento de Matemática, 70.910-900
Brasília, DF, Brazil
email: ruviaro@unb.br

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