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Minimax Theory and its Applications 04 (2019), No. 1, 001--019
Copyright Heldermann Verlag 2019

On a Fractional p&q Laplacian Problem with Critical Growth

Vincenzo Ambrosio
Department of Mathematics, EPFL SB CAMA, Station 8, 1015 Lausanne, Switzerland
vincenzo.ambrosio2@unina.it

Teresa Isernia
Dip. di Ingegneria Industriale e Scienze Matematiche, Univ. Politecnica delle Marche, Via Brecce Bianche 12, 60131 Ancona, Italy
teresa.isernia@unina.it

Gaetano Siciliano
Departamento de Matemática, Universidade de Sao Paulo, Rua do Matao 1010, 05508-090 Sao Paulo, Brazil
sicilian@ime.usp.br


[Abstract-pdf]

\newcommand{\q}{q^{*}_{s}} \newcommand{\R}{\mathbb R} We deal with a class of nonlocal problems of the type \begin{equation*} \left\{ \begin{array}{ll} (-\Delta)^{s}_{p}u+(-\Delta)^{s}_{q}u= \lambda |u|^{r-2}u+|u|^{\q-2}u &\mbox{ in } \Omega\\[1mm] u=0 &\mbox{ in } \R^{N}\setminus \Omega, \end{array} \right. \end{equation*} where $s\in (0, 1)$, $10$ is a parameter. Roughly speaking, when $r$ is ``large'' we prove the existence of a solution for large values of $\lambda$ and when $r$ is ``small'' we prove the existence of infinitely many solutions for small values of $\lambda$.

Keywords: Fractional Laplacians, variational methods, critical exponent.

MSC: 47G20, 35R11, 35A15.

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