On the Choquard equation with double critical Sobolev exponents in ℝN

In this paper, we study the existence and nonexistence results to the Choquard equation −Δu=(∫RN|u(y)|2α∗|x−y|αdy)|u|2α∗−2u±|u|q−2uinRN, where 2α∗=2N−αN−2, 0<α<N, 1<q≤2∗, 2∗=2NN−2, N≥3. We first use the Pohozaev-type identity to show the nonexistence of solutions for 1<q<2∗. When the equation has double critical exponents, i.e. q=2∗, we obtain the existence of radial ground state solutions by the Nehari manifold and Mountain pass theorem.