We consider a semilinear elliptic problemwhere Iα is a Riesz potential and p > 1. This family of equations includes the Choquard or nonlinear Schrödinger-Newton equation. For an optimal range of parameters we prove the existence of a positive groundstate solution of the equation. We also establish regularity and positivity of the groundstates and prove that all positive groundstates are radially symmetric and monotone decaying about some point. Finally, we derive the decay asymptotics at infinity of the groundstates.
Abstract. We prove the existence of a nontrivial solution u ∈ H 1 (R N ) to the nonlinear Choquard equationwhere Iα is a Riesz potential, under almost necessary conditions on the nonlinearity F in the spirit of Berestycki and Lions. This solution is a groundstate; if moreover F is even and monotone on (0, ∞), then u is of constant sign and radially symmetric.
Abstract. We survey old and recent results dealing with the existence and properties of solutions to the Choquard type equationsand some of its variants and extensions.
Mathematics Subject Classification (2010). 35Q55 (35R09, 35J91).
Abstract. We study the nonlocal Schrödinger-Poisson-Slater type equationwhere N ∈ N, p > 1, q > 1 and Iα is the Riesz potential of order α ∈ (0, N ). We introduce and study the Coulomb-Sobolev function space which is natural for the energy functional of the problem and we establish a family of associated optimal interpolation inequalities. We prove existence of optimizers for the inequalities, which implies the existence of solutions to the equation for a certain range of the parameters. We also study regularity and some qualitative properties of solutions. Finally, we derive radial Strauss type estimates and use them to prove the existence of radial solutions to the equation in a range of parameters which is in general wider than the range of existence parameters obtained via interpolation inequalities.
Abstract. We consider nonlinear Choquard equationis an external potential and Iα(x) is the Riesz potential of order α ∈ (0, N ). The power α N + 1 in the nonlocal part of the equation is critical with respect to the Hardy-Littlewood-Sobolev inequality. As a consequence, in the associated minimization problem a loss of compactness may occur. We prove that ifthen the equation has a nontrivial solution. We also discuss some necessary conditions for the existence of a solution. Our considerations are based on a concentration compactness argument and a nonlocal version of Brezis-Lieb lemma.
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