Existence and qualitative properties of solutions for Choquard equations with a local term

In this paper, a nontrivial solution u is an element of H-1 (R-N) to the autonomous Choquard equation with a local term - Delta u + lambda u = (I-alpha * vertical bar u vertical bar(p)) vertical bar u vertical bar(p-2)u + vertical bar u vertical bar(q-2)u in R-N is obtained, where N >= 3, alpha is an element of (0, N), lambda > 0 is a constant, I-alpha is the Riesz potential, N+alpha/N < p < N+alpha/N-2 and q is an element of (2,2* = 2N/N-2 ). Under some further assumptions on p and q, the regularity and the Poholaev identity of the solution are established, and then it is shown that the obtained solution is a groundstate of mountain pass type. Moreover, the positivity and symmetry of the groundstate are also considered. By using the results obtained for the autonomous equation, a positive groundstate solution for the nonautonomous equation - Delta u + V(x)u = (I-alpha * vertical bar u vertical bar(p)) vertical bar u vertical bar(p-2)u + vertical bar u vertical bar(q-2)u in R-N is also found under some assumptions on V(x). (C) 2018 Elsevier Ltd. All rights reserved.