12-61 Alessio Fiscella, Raffaella Servadei, Enrico Valdinoci

A resonance problem for non-local elliptic operators (54K, LaTeX) May 18, 12

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Abstract. In this paper we consider a resonance problem driven by a non-local integrodifferential operator $\mathcal L_K$ with homogeneous Dirichlet boundary conditions. This problem has a variational structure and we find a solution for it using the Saddle Point Theorem. We prove this result for a general integrodifferential operator of fractional type and from this, as a particular case, we derive an existence theorem for the following fractional Laplacian equation $$ \left\{ egin{array}{ll} (-\Delta)^s u=\lambda a(x)u+f(x,u) & {\mbox{ in }} \Omega\ u=0 & {\mbox{ in }} \mathbb{R}^n\setminus \Omega\,, \end{array} ight.$$ when $\lambda$ is an eigenvalue of the related non-homogenous linear problem with homogeneous Dirichlet boundary data.

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