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Zhang Binlin, Giovanni Molica Bisci and Raffaella Servadei
Superlinear nonlocal fractional problems with infinitely many solutions
(514K, pdf)

ABSTRACT.  In this paper we study the existence of infinitely many weak solutions for equations driven by nonlocal integrodifferential operators with homogeneous Dirichlet boundary conditions. A model for these operators is given by the fractional Laplacian 
$$-(-\Delta)^s u(x):= 
\int_{\RR^n}rac{u(x+y)+u(x-y)-2u(x)}{|y|^{n+2s}}\,dy\,, 
\,\,\,\,\, x\in \RR^n\,$$ 
where $s\in (0,1)$ is fixed. 
We consider different superlinear growth assumptions on the nonlinearity, starting from the well-known Ambrosetti--Rabinowitz condition. In this framework we obtain three different results about the existence of infinitely many weak solutions for the problem under consideration, by using the Fountain Theorem. 
All these theorems extend some classical results for semilinear Laplacian equations to the nonlocal fractional setting.