EXISTENCE OF WEAK SOLUTIONS TO A CLASS OF SCHRODINGER TYPE EQUATIONS INVOLVING THE FRACTIONAL p-LAPLACIAN IN R-N

Abstract

We are concerned with the following elliptic equations: (-Delta)(p)(s)u + V(x)vertical bar u vertical bar(p-2)u = lambda g(x,u) in R-N, where (-Delta)(p)(s) is the fractional p-Laplacian operator with 0 < s < 1 < p < + infinity, sp < N, the potential function V : R-N -> (0, infinity) is a continuous potential function, and g : R-N x R -> R satisfies a Caratheodory condition. We show the existence of at least one weak solution for the problem above without the Ambrosetti and Rabinowitz condition. Moreover, we give a positive interval of the parameter lambda for which the problem admits at least one nontrivial weak solution when the nonlinearity g has the subcritical growth condition.