Existence and non-existence results for Kirchhoff-type problems with convolution nonlinearity

Abstract

This paper is concerned with the following Kirchhoff-type problem with convolution nonlinearity: -(a + b R3 |u|2 dx)u + V(x)u = (Ia∗F(u))f(u), x R3, u H1(R3), where a, b > 0, Ia : R3 R, with a (0, 3), is the Riesz potential, V C(R3, [0, 8)), f C(R, R) and F(t) = t 0 f(s) ds. By using variational and some new analytical techniques, we prove that the above problem admits ground state solutions under mild assumptions on V and f . Moreover, we give a non-existence result. In particular, our results extend and improve the existing ones, and fill a gap in the case wheref(u) = |u|q-2u, with q (1 + a/3, 2].