Existence and qualitative properties of solutions for Choquard equations with a local term. (February 2019) - Li, Xinfu; Ma, Shiwang; Zhang, Guang
Abstract: In this paper, a nontrivial solution u ∈ H 1 ( R N ) to the autonomous Choquard equation with a local term − Δ u + λ u = ( I α ∗ | u | p ) | u | p − 2 u + | u | q − 2 u in R N is obtained, where N ≥ 3, α ∈ ( 0, N ), λ > 0 is a constant, I α is the Riesz potential, N + α N < p < N + α N − 2 and q ∈ ( 2, 2 ∗ = 2 N N − 2 ) . Under some further assumptions on p and q, the regularity and the Pohožaev 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 − Δ u + V ( x ) u = ( I α ∗ | u | p ) | u | p − 2 u + | u | q − 2 u in R N is also found under some assumptions on V ( x ) .