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Li, Xinfu; Ma, Shiwang; Zhang, Guang
Nonlinear analysis: real world applications, February 2019, 2019-02-00, 20190201, Volume: 45Journal Article
In this paper, a nontrivial solution u∈H1(RN) to the autonomous Choquard equation with a local term −Δu+λu=(Iα∗|u|p)|u|p−2u+|u|q−2uinRNis obtained, where N≥3, α∈(0,N), λ>0 is a constant, Iα is the Riesz potential, N+αN<p<N+αN−2 and q∈(2,2∗=2NN−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 −Δu+V(x)u=(Iα∗|u|p)|u|p−2u+|u|q−2uinRNis also found under some assumptions on V(x).
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