Existence and multiplicity of solutions for fractional Schrodinger-p-Kirchhoff equations in RN

This paper concerns the existence and multiplicity of solutions for a nonlinear Schrodinger-Kirchhoff-type equation involving the fractional p-Laplace operator in R-N. Precisely, we study the Kirchhoff-type problem (a+b integral integral(R2N)vertical bar u(x) - u(y)vertical bar(p) \ vertical bar x-y vertical bar(N+sp) dx dy) (-Delta)(p)(s)u + V(X)vertical bar u vertical bar(p-2) u=f(x,u) in R-N, where a, b > 0, (-Delta)(s)(p) is the fractional p-Laplacian with 0 < s < 1 < p < N\s, V : R-N -> R and f : R-N x R -> R are continuous functions while V can have negative values and f fulfills suitable growth assumptions. According to the interaction between the attenuation of the potential at infinity and the behavior of the nonlinear term at the origin, using a penalization argument along with L-infinity-estimates and variational methods, we prove the existence of a positive solution. In addition, we also establish the existence of infinitely many solutions provided the nonlinear term is odd.