Ground state solutions for the fractional problems with dipole-type potential and critical exponent

Abstract

We are concerned with ground state solutions of the fractional problems with dipole-type potential and critical exponent. Under certain conditions on the dipole-type potential and the parameter, we show that the structure of the Palais-Smale sequence goes to zero weakly, and establish the existence of ground state solution to the above problems by using a new analytical method not involving the concentration-compactness principle.

Some recent developments on fractional parabolic equations

“…In [21], a direct method of moving planes on fractional equations was introduced to obtain symmetry and nonexistence of solutions for nonlinear equations involving the fractional Laplacian on various domains. This method has become a powerful tool in investigating qualitative properties of fractional equations and has been applied by numerous researchers to solve a wide variety of problems (see [9,11,17,18,19,20,21,27,28,12,29,35,36,42,43,56,68,69,73,71,75,76,80,79,91,94,96,102,103,101] and the references therein). This idea was then modified in [19] [20] to study equations involving fully nonlinear nonlocal operators and degenerate nonlinear nonlocal operators such as fractional p-Laplacians, and obtained symmetry, monotonicity, and nonexistence of solutions.…”