Existence and multiplicity results for Pucci’s operators involving nonlinearities with zeros

Abstract

We study the problem

$$ \left\{\begin{array}{ll} {-\varepsilon^{2}\mathcal{M}^+_{\lambda,\Lambda}(D^{2}u) = f (x, u)} \quad\; {\rm in} \; \Omega,\\ {u = 0} \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad {\rm on} \; \partial{\Omega}, \end{array} \right.$$

where Ω is a smooth bounded domain in \({\mathbb{R}^{N},N > 2,}\) and show it possesses nontrivial solutions for small values of ε provided f is a nonnegative continuous function which has a positive zero. The multiplicity result is based on degree theory together with a new Liouville type theorem for \({-{M}^+_{\lambda,\Lambda}(D^{2}u) = f(u)}\) in \({\mathbb{R}^{N}}\) for nonnegative nonlinearities with zeros.

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Correspondence to Salomón Alarcón.

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Communicated by A. Malchiodi.

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Alarcón, S., Iturriaga, L. & Quaas, A. Existence and multiplicity results for Pucci’s operators involving nonlinearities with zeros. Calc. Var. 45, 443–454 (2012). https://doi.org/10.1007/s00526-011-0465-0

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Mathematics Subject Classification (2000)

  • 35J60
  • 35B40
  • 35B45
  • 35B50
  • 35D05
  • 49L25