Existence and uniqueness results for possibly singular nonlinear elliptic equations with measure data

  • Linda Maria De Cave
  • Riccardo Durastanti
  • Francescantonio Oliva


We study existence and uniqueness of solutions to a nonlinear elliptic boundary value problem with a general, and possibly singular, lower order term, whose model is
$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\Delta _p u = H(u)\mu &{}\quad \text {in}\ \Omega ,\\ u>0 &{}\quad \text {in}\ \Omega ,\\ u=0 &{}\quad \text {on}\ \partial \Omega . \end{array}\right. } \end{aligned}$$
Here \(\Omega \) is an open bounded subset of \({\mathbb {R}}^N\) (\(N\ge 2\)), \(\Delta _p u:= {\text {div}}(|\nabla u|^{p-2}\nabla u)\) (\(1<p<N\)) is the p-laplacian operator, \(\mu \) is a nonnegative bounded Radon measure on \(\Omega \) and H(s) is a continuous, positive and finite function outside the origin which grows at most as \(s^{-\gamma }\), with \(\gamma \ge 0\), near zero.


Nonlinear elliptic equations Singular elliptic equations Measure data 

Mathematics Subject Classification

35J60 35J61 35J75 35R06 


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Linda Maria De Cave
    • 1
  • Riccardo Durastanti
    • 2
  • Francescantonio Oliva
    • 2
  1. 1.Institut für MathematikUniversität ZürichZurichSwitzerland
  2. 2.Dipartimento di Scienze di Base e Applicate per l’ Ingegneria“Sapienza” Università di RomaRomeItaly

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