12-145 Giovanni Molica Bisci and Raffaella Servadei

A bifurcation result for non-local fractional equations (329K, pdf) Nov 26, 12

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Abstract. In the present paper we consider problems modeled by the following non-local fractional equation $$\left\{ egin{array}{ll} (-\Delta)^s u-\lambda u=\mu f(x,u) & {\mbox{ in }} \Omega\ u=0 & {\mbox{ in }} \RR^n\setminus \Omega\,, \end{array} ight.$$ where $s\in (0,1)$ is fixed, $(-\Delta )^s$ is the fractional Laplace operator, $\lambda$ and $\mu$ are real parameters, $\Omega$ is an open bounded subset of $\RR^n$, $n>2s$\,, with Lipschitz boundary and $f$ is a function satisfying suitable regularity and growth conditions. A critical point result for differentiable functionals is exploited, in order to prove that the problem admits at least one non-trivial and non-negative (non-positive) solution, provided the parameters $\lambda$ and $\mu$ lie in a suitable range. The existence result obtained in the present paper may be seen as a bifurcation theorem, which extends some results, well known in the classical Laplace setting, to the non-local fractional framework.

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