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2017 Blow-up solutions for some nonlinear elliptic equations involving a Finsler-Laplacian
Francesco Della Pietra, Giuseppina di Blasio
Publ. Mat. 61(1): 213-238 (2017). DOI: 10.5565/PUBLMAT_61117_08

Abstract

In this paper we prove existence results and asymptotic behavior for strong solutions $u\in W^{2,2}_{\operatorname{loc}}(\Omega)$ of the nonlinear elliptic problem \begin{equation}\label{abstr} \begin{cases} -\Delta_{H}u+H(\nabla u)^{q}+\lambda u=f&\text{in }\Omega,\\ u\rightarrow +\infty &\text{on }\partial\Omega, \end{cases}\tag{P} \end{equation} where $H$ is a suitable norm of $\mathbb R^{n}$, $\Omega\subset\mathbb R^{n}$ is a bounded domain, $\Delta_{H}$ is the Finsler Laplacian, $1\lt q\le 2$, $\lambda>0$, and $f$ is a suitable function in $L^{\infty}_{\operatorname{loc}}$. Furthermore, we are interested in the behavior of the solutions when $\lambda\rightarrow 0^{+}$, studying the so-called ergodic problem associated to (P). A key role in order to study the ergodic problem will be played by local gradient estimates for (P).

Citation

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Francesco Della Pietra. Giuseppina di Blasio. "Blow-up solutions for some nonlinear elliptic equations involving a Finsler-Laplacian." Publ. Mat. 61 (1) 213 - 238, 2017. https://doi.org/10.5565/PUBLMAT_61117_08

Information

Received: 3 June 2015; Revised: 29 October 2015; Published: 2017
First available in Project Euclid: 22 December 2016

zbMATH: 1366.35034
MathSciNet: MR3590120
Digital Object Identifier: 10.5565/PUBLMAT_61117_08

Subjects:
Primary: 35B44, 35J25, 35J60

Rights: Copyright © 2017 Universitat Autònoma de Barcelona, Departament de Matemàtiques

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Vol.61 • No. 1 • 2017
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