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Blow-up solutions for some nonlinear elliptic equations involving a Finsler-Laplacian

Francesco Della Pietra and Giuseppina di Blasio

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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).

Article information

Publ. Mat., Volume 61, Number 1 (2017), 213-238.

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J60: Nonlinear elliptic equations 35J25: Boundary value problems for second-order elliptic equations 35B44: Blow-up

Anisotropic elliptic problems Finsler Laplacian Blow-up solutions


Della Pietra, Francesco; di Blasio, Giuseppina. Blow-up solutions for some nonlinear elliptic equations involving a Finsler-Laplacian. Publ. Mat. 61 (2017), no. 1, 213--238. doi:10.5565/PUBLMAT_61117_08.

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