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A singular elliptic equation with natural growth in the gradient and a variable exponent

  • José Carmona
  • Pedro J. Martínez-Aparicio
  • Julio D. RossiEmail author
Article

Abstract

In this paper we consider singular quasilinear elliptic equations with quadratic gradient and a singular term with a variable exponent
$$\begin{cases} -\Delta u + \frac{|{\nabla}u|^2}{u^{\gamma(x)}} = f & {\rm in} \, \Omega \\ u = 0 & {\rm on} \, \partial \Omega \end{cases}$$
Here \({\Omega}\) is an open bounded set of \({\mathbb{R}^N}\), \({\gamma(x)}\) is a positive continuous function and f is positive function that belongs to a certain Lebesgue space. We show, among other results, that there exists a solution in the natural energy space \({H^1_0 (\Omega)}\) to this problem when \({\gamma (x)}\) is strictly less than 2 in a strip around the boundary; while there is no solution in the energy space when there exists \({\Gamma \subset \partial \Omega}\) with \({|\Gamma|_{N-1} > 0}\) such that \({\gamma(x) > 2}\) on \({\Gamma}\). Moreover, since we work by approximation we can analyze the behavior of the approximated solutions \({u_n}\) in the case in which there is no solution in \({H_0^1(\Omega)}\).

Keywords

Nonlinear elliptic equations Singular natural growth gradient terms Positive solutions Variable exponent 

Mathematics Subject Classification

35A01 35B09 35B45 35D30 35J25 35J60 35J75 

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Copyright information

© Springer Basel 2015

Authors and Affiliations

  • José Carmona
    • 1
  • Pedro J. Martínez-Aparicio
    • 2
  • Julio D. Rossi
    • 3
    Email author
  1. 1.Departamento de MatemáticasUniversidad de AlmeríaAlmeríaSpain
  2. 2.Departamento de Matemática Aplicada y Estadística, Campus Alfonso XIIIUniversidad Politécnica de CartagenaMurciaSpain
  3. 3.Departamento de MatemáticaFCEyN UBA, Ciudad UniversitariaBuenos AiresArgentina

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