Solitary waves for nonlinear Klein–Gordon–Maxwell and Schrödinger–Maxwell equations

Abstract

In this paper we study the existence of radially symmetric solitary waves for nonlinear Klein-Gordon equations and nonlinear Schrödinger equations coupled with Maxwell equations. The method relies on a variational approach and the solutions are obtained as mountain-pass critical points for the associated energy functional.

Multiple solutions for nonhomogeneous Schrödinger–Maxwell and Klein– Gordon–Maxwell equations on R 3

“…Let us recall some recent results in the literature of nonlinear Schrödinger-Maxwell equations (1). The case of h ≡ 0, that is the homogeneous case, has been studied widely in [4,[14][15][16][17]20,22,24,25]) when V is a constant or radially symmetric, and in [27,29] when V is not radially symmetric. Very recently, Azzollni and Pomponio in [1] proved the existence of a ground state solution (namely for solution which minimizes the action functional among all the solutions) for system (1) with f (x, u) = |u| s−2 u (4 < s < 6) and non-constant potential V which may be unbounded from below; Zhao and Zhao [28] established the existence of a positive solution for problem (1) with a critical Sobolev exponent and constant potential V ; Chen and Tang [13] obtained the existence of infinitely many large energy solutions for system (1) with f (x, u) satisfying Amborosetti-Rabinowitz type condition [see (f2)] and V being nonradially symmetric.…”

“…In recent years, some existence and nonexistence results for the KleinGordon-Maxwell equations (2) have been proved in the case f (x, u) = |u| q−2 u and h(x) ≡ 0. In [6,7] the authors found the existence of infinitely many radially symmetric solutions (u, φ) in H 1 (R 3 ) × D 1,2 (R 3 ) for system (2) when 4 < q < 6; in [17] the range q ∈ (2, 4] was also covered; in [2] the existence of a ground state solution (u, φ) ∈ H 1 (R 3 ) × D 1,2 (R 3 ) for problem (2) was get when 4 ≤ q < 6 and m > ω or 2 < q ≤ 4 and m √ q − 2 > ω √ 6 − q; in [16], the nonexistence results for system (2) were obtained for q ≤ 2 or q ≥ 6, respectively. For the case f (x, u) = λ|u| q−2 u + |u| 2 * −2 u and h ≡ 0, Cassani in [12] get that problem (2) has at least a radially symmetric (nontrivial) solu-…”

Multiple positive solutions for the nonlinear Schrödinger–Poisson system

“…In recent years, there has been increasing interest in studying problem (1.1). In the case of V (x) ≡ 1, K(x) ≡ λ > 0, the existence of radially symmetric positive solutions of system (1.1) was obtained by D'Aprile and Mugnai in [9] and Ruiz in [20] for p ∈ (3, 6). Azzollini and Pomponio in [4] established the existence of ground state solutions for p ∈ (3, 6).…”

Quantitative properties on the steady states to a Schrödinger–Poisson–Slater System

“…[2,15,17] and the reference therein), the following Schrödinger-Poisson-Slater (SPS) system in terms of the wave function ψ : R 3 × [0, T ) → C      i ∂ψ ∂t = −∆ x ψ + V (x, t)ψ + C S |ψ(x, t)| 2α ψ, lim x→∞ ψ(x, t) → 0 ψ(x, t = 0) = ψ 0 (x), −∆ x V = ǫ|ψ| 2 , lim x→∞ V (x, t) → 0 has been studied extensively in recent years, see [1,2,3,6,17,20,21] for instance. Here α, ǫ, C S are constants., such as ǫ = +1 (repulsive case), ǫ = −1 (attractive case) depending on the type of interaction considered.…”

“…which is a special case of Schrödinger-Maxwell equations ( [6]). The existence of standing waves has been studied from various perspectives in the vast mathematical literature.…”