| Abstract: |
| In this talk we consider a nonlinear Dirichlet problems driven by the fractional Laplacian. We prove the existence of two nontrivial solutions, exploiting a two nontrivial critical points theorem for differentiable functionals. Moreover, we study such Dirichlet problems when the nonlinearity has a critical growth, and we prove the existence of one positive weak solution, applying a local minimum theorem for energy functionals that satisfy a suitable type of Palais Smale condition.
This is a joint work with Antonio Iannizzotto.
\bibitem{label}
G. Bonanno, G. D`Agu\`{i} and D. O`Regan, A local minimum theorem and critical nonlinearities, {\em Analele Universitatii ``Ovidius`` Constanta-Seria Matematica}
\textbf{24} (2016), 67--86.
\bibitem{label}
G. Bonanno, G. D`Agu\`{i}, Two non-zero solutions for elliptic Dirichlet problems, {\em Zeitschrift fuer Analysis und Ihre Anwendungen} \textbf{35} (2016), 449--465.
\bibitem{label}
S. Frassu, A. Iannizzotto, Positive solutions for the fractional Laplacian with subcritical and critical nonlinearities, preprint. |
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