Normalized solutions for Choquard equations with general nonlinearities

Abstract

In this paper, we prove the existence of positive solutions with prescribed L 2-norm to the following Choquard equation: −∆u − λu = (Iα * F (u))f (u), x ∈ R 3 , where λ ∈ R, α ∈ (0, 3) and Iα : R 3 → R is the Riesz potential. Under the weaker conditions, by using a minimax procedure and some new analytical techniques, we show that for any c > 0, the above equation possesses at least a couple of weak solution (ūc,λc) ∈ Sc × R − such that ūc 2 2 = c.

Nonexistence, existence and symmetry of normalized ground states to Choquard equations with a local perturbation

“…under a set of assumptions on f , which when f takes the special form f (s) = C 1 |s| r−2 s + C 2 |s| p−2 s requires that p * < r ≤ p < p. Bartsch, Liu and Liu [1] further considered the existence of normalized ground state and the existence of infinitely many normalized solutions to (1.5) in all dimensions N ≥ 1. Recently, Yuan, Chen and Tang [39] reconsidered (1.5) with more general f ∈ C(R, R). When p = p * and 2 < q < q * , [23] considered the existence and orbital stability of the normalized ground state to (1.1).…”

Standing waves to upper critical Choquard equation with a local perturbation: multiplicity, qualitative properties and stability

“…For p = p * , by scaling invariance, the result is delicate, see [7] and [41] for details. See [1,21,42] for studies to Choquard equation with general nonlinearity. For (1.5) with p = p, Moroz and Van Schaftingen [28] showed that (1.5) has no solutions in H 1 (R N ) for fixed λ < 0.…”

Normalized solutions for a Choquard equation with exponential growth in $$\mathbb {R}^{2}$$

“…The authors in [14] used a minimax procedure and the concentration compactness to show equation (1.5) has at least a weak solution. For N = 3, Yuan et al [30] complemented and generalized the known results in [14]. In [5], Bartsch et al proved the existence of a least energy solution of equation (1.5) in all dimensions N ≥ 1, which is simpler and more transparent than the one from [14].…”