Ground states and multiple solutions for Choquard-Pekar equations with indefinite potential and general nonlinearity

Strongly indefinite Choquard equation in ℝ2 with critical exponential growth

“…Remark 1.3. Theorem 1.2 extends and improves the related results in previous works 7,12,27,28,31 involving the polynomial growth nonlinearity and results in Alves et al 24, Theorem 1.3 and Yang 25 which concern the positive potential case, and complements the result in Qin and Tang 32, Theorem 1.3 for the indefinite potential case.…”

“…For the indefinite case, that is, , using some general assumptions on the nonlinearity and assuming that is large at infinity, Qin et al 12 studied a more general type equation of () with and obtained the existence of ground state solutions and infinitely many solutions via the symmetric mountain pass theorem. In this paper, we are interested in the strongly indefinite case; in particular, the potential satisfies is 1‐periodic in and , and 0 lies in a gap of the spectrum , that is, …”

“…For the indefinite case, that is, 𝜎(− △ +V) ∩ (−∞, 0) ≠ ∅, using some general assumptions on the nonlinearity 𝑓 and assuming that V is large at infinity, Qin et al 12 studied a more general type equation of (1.1) with N ≥ 3 and obtained the existence of ground state solutions and infinitely many solutions via the symmetric mountain pass theorem. In this paper, we are interested in the strongly indefinite case; in particular, the potential V satisfies…”

“…When the spectrum , that is, the positive definite situation which includes the case and the case that the potential is nonnegative and vanishes on an open bounded domain in , Equation () with has been extensively studied in recent years via the mountain pass theorem, Ekeland's variational principle, Strauss's lemma and the method of moving plane, and so forth; see previous works for the existence, multiplicity, and uniqueness results, 3,5,7–14 for the sign‐changing solutions, 15–17 and for the semiclassical solutions 18–21 . In recent review article, 22 a guide to the Choquard equation was given by Moroz and Schaftingen.…”

“…12),(3.42),(3.43), and Qin and Tang, 32, Lemma 4.8 we derive that⟨Φ ′ (ṽ), 𝜙⟩ = ∫ R 2 (∇ṽ • ∇𝜙 + V(x)ṽ𝜙) dx − ∫ R 2 [I 𝛼 * F(ṽ)]𝑓 (ṽ)𝜙dx = lim n→∞ { ∫ R 2 (∇ṽ n ∇𝜙 + V(x)ṽ n 𝜙) dx − ∫ R 2 [I 𝛼 * F(ṽ n )]𝑓 (ṽ n )𝜙dx } = lim n→∞ [ ⟨Φ ′ (ṽ n ), 𝜙 n ⟩ − ∫ R 2 [I 𝛼 * F(ṽ n )]𝑓 (ṽ n ) φn dx ] = − lim n→∞ ∫ R 2 [I 𝛼 * F(ṽ n )]𝑓 (ṽ n ) φn dx = 0.…”