Existence and multiplicity of solutions for fractional Schrodinger-Kirchhoff equations with Trudinger-Moser nonlinearity

We study the existence and multiplicity of solutions for a class of fractional Schrodinger-Kirchhoff type equations with the Trudinger-Moser nonlinearity. More precisely, we consider {M(parallel to u parallel to(N/s)) [(-Delta)(N/s)(s)u + V(x)vertical bar u vertical bar(N/s - 1)u] = f(x, u) + lambda h(x)vertical bar u vertical bar(p-2)u in R-N, parallel to u parallel to = (integral integral(R2N) vertical bar u(x)-u(y)vertical bar(N/s)/vertical bar x-y vertical bar(2N)dxdy + integral(RN) V(x)vertical bar u vertical bar(N/s)dx)(s/N), where M : [0, infinity] -> [0, infinity) is a continuous function, s is an element of(0, 1), N >= 2, lambda > 0 is a parameter, 1 < p < infinity, (-Delta)(N/s)(s) is the fractional N/s-Laplacian, V : R-N -> (0, infinity) is a continuous function, f : R-N x R -> R is a continuous function, and h : R-N -> [0, infinity) is a measurable function. First, using the mountain pass theorem, a nonnegative solution is obtained when f satisfies exponential growth conditions and lambda is large enough, and we prove that the solution converges to zero in W-V(s,N/s) (R-N) as lambda -> infinity. Then, using the Ekeland variational principle, a nonnegative nontrivial solution is obtained when lambda is small enough, and we show that the solution converges to zero in W-V(s,N/s) as lambda -> 0. Furthermore, using the genus theory, infinitely many solutions are obtained when M is a special function and lambda is small enough. We note that our paper covers a novel feature of Kirchhoff problems, that is, the Kirchhoff function M(0) = 0. (C) 2018 Elsevier Ltd. All rights reserved.