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Existence of solution for quasilinear equations involving local conditions

Part of: Partial differential equations Elliptic equations and systems

Published online by Cambridge University Press:  17 September 2019

Patricio Cerda  and
Leonelo Iturriaga
Patricio Cerda
Affiliation:
Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile (patricio.cerda@usach.cl)
Leonelo Iturriaga
Affiliation:
Departamento de Matemática, Universidad Técnica Federico Santa María, Casilla V-110, Avda. España, 1680, Valparaíso, Chile (leonelo.iturriaga@usm.cl)
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Abstract

In this paper, we study the existence of weak solutions of the quasilinear equation

\begin{cases} -{\rm div} (a(\vert \nabla u \vert ^2)\nabla u)=\lambda f(x,u) &{\rm in} \ \Omega,\\ u=0 &{\rm on} \ \partial\Omega, \end{cases}
where a : ℝ → [0, ∞) is C1 and a nonincreasing continuous function near the origin, the nonlinear term f : Ω × ℝ → ℝ is a Carathéodory function verifying certain superlinear conditions only at zero, and λ is a positive parameter. The existence of the solution relies on C1-estimates and variational arguments.

Keywords

Existence solutionquasilinear equationsvariational methods

MSC classification

Primary: 35J93: Quasilinear elliptic equations with mean curvature operator
Secondary: 35B09: Positive solutions 35J50: Variational methods for elliptic systems 35J62: Quasilinear elliptic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 6 , December 2020 , pp. 3074 - 3086
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Copyright
Copyright © 2019 The Royal Society of Edinburgh

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