On the semilinear fractional elliptic equations with singular weight functions

Abstract

In the paper, we study a class of semilinear fractional semilinear elliptic equations involving concave-convex nonlinearities: (−∆) α u + V λ (x) u = f (x) |u| q−2 u + g (x) |u| p−2 u in R N , u ∈ H α (R N), where α ∈ (0, 1], 1 < q < 2 < p < 2 * α 2 * α = 2N N −2α for N > 2α , the potential V λ (x) = λa(x) − b(x) and the parameter λ > 0. Under some suitable assumptions on a, b and the weight functions f, g, we obtain the existence and multiplicity of non-trivial (positive) solutions for λ large enough. An interesting phenomenon is that we do not need the condition that weight functions f, g are integrable or bounded on whole space R N .