 12145 Giovanni Molica Bisci and Raffaella Servadei
 A bifurcation result for nonlocal fractional equations
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Nov 26, 12

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Abstract. In the present paper we consider problems modeled by the following nonlocal 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 nontrivial and nonnegative (nonpositive) 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 nonlocal fractional framework.
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