Infinitely Many Solutions for the Fractional Nonlinear Schrödinger Equations of a New Type

Abstract

. This paper, we study the multiplicity of solutions for the fractional Schr¨odinger equation with s ∈ ( 0,1 ) , N ≥ 3, p ∈ ( 1, 2 N N − 2 s − 1 ) and lim | y |→ + ∞ V ( y ) > 0. By assuming suitable decay property of the radial potential V ( y ) = V ( | y | ) , we construct another type of solutions concentrating at infinite vertices of two similar equilateral polygonal with infinitely large length of sides. Hence, besides the length of each polygonal, we must consider one more parameter, that is the height of the podetium, simultaneously. Another difficulty lies in the non-local property of the operator ( − ∆ ) s and the algebraic decay involving the approximation solutions make the estimates become more subtle.