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Existence of positive solutions for a class of \( p \& q\) elliptic problem with critical exponent and discontinuous nonlinearity

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Abstract

In this paper we study the existence of positive solutions to a class of \( p \& q\) elliptic problems given by

$$\begin{aligned} -\text{ div }(a(|\nabla u|^{p})|\nabla u|^{p-2}\nabla u)= f(u) + |u|^{q^{*}-2}u \ \text{ in } \ \Omega , \ \ u=0 \ \text{ on } \ \ \partial \Omega , \end{aligned}$$

where \(\Omega \subset {\mathbb {R}}^{N}\) is bounded, \(2 \le p \le q< q^{*}\), \(f:{\mathbb {R}}\rightarrow {\mathbb {R}}\) is a function that can have an uncountable set of discontinuity points and the function a is a continuous function. This result to extend previous ones to a larger class of \( p \& q\) type problems.

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Authors and Affiliations

  1. Departamento de Matemática, Universidade de Brasilia - UNB, Brasília, DF, CEP. 70910-900, Brazil

    Giovany M. Figueiredo

  2. Faculdade de Matemática, Universidade Federal do Pará - UFPA, Belém, PA, CEP. 66075-110, Brazil

    Rúbia G. Nascimento

Authors
  1. Giovany M. Figueiredo
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  2. Rúbia G. Nascimento
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Correspondence to Giovany M. Figueiredo.

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Communicated by A. Constantin.

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Figueiredo, G.M., Nascimento, R.G. Existence of positive solutions for a class of \( p \& q\) elliptic problem with critical exponent and discontinuous nonlinearity. Monatsh Math 189, 75–89 (2019). https://doi.org/10.1007/s00605-018-1200-0

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