Journal of the
Korean Mathematical Society JKMS

We are concerned with elliptic equations in $\mathbb R^N$, driven by a non-local integro-differential operator, which involves the fractional Laplacian. The main aim of this paper is to prove the existence of small solutions for our problem with negative energy in the sense that the sequence of solutions converges to $0$ in the $L^{\infty}$-norm by employing the regularity type result on the $L^{\infty}$-boundedness of solutions and the modified functional method.

Keywords: integrodifferential operators, fractional Laplacian, variational methods, infinitely many solutions

MSC numbers: 35R11, 47G20, 35A15, 58E30