Infinitely many solutions for a class of elliptic problems involving the fractional Laplacian

In this paper, we study the existence of nontrivial solution to a quasi-linear problem (-Delta)(s)(p)u + V (x) vertical bar u vertical bar(p-2)u = Q(x) f (x, u), x is an element of R-N, where (-Delta)(s)(p)u(x) = 2 lim is an element of -> 0 integral R-N backslash B epsilon(x) vertical bar u(x) - u(y) vertical bar (p-2) (u(x) - u(y)) / vertical bar x - y vertical bar (N+sp) dy, x epsilon R-N is a nonlocal and nonlinear operator and p is an element of (1, infinity), s is an element of (0, 1). We study two cases: if f (x, u) is sublinear, then we get infinitely many solutions for (P) by Clark's theorem; if f (x, u) is superlinear, we obtain infinitely many solutions of the problem (P) by symmetric mountain pass theorem.