A multiplicity result for the Schrodinger–Maxwell equations with negative potential

Abstract

Abstract.We prove the existence of a sequence of radial solutions with negative energy of the Schrödinger-Maxwell equations under the action of a negative potential. Introduction.In this paper we study the interaction between the electromagnetic field and the wave function related to a quantum nonrelativistic charged particle, which is described by the Schrödinger equation.In [2, 3, 11] the case in which the electromagnetic field is given has been studied. Here we shall assume that the unknowns of the problem are both the wave function ψ = ψ(x, t) and the gauge potentials ϕ = ϕ(x, t) and A = A(x, t) related to the electromagnetic fields E, H by the equations

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.…”

Ground states for asymptotically periodic Schrödinger-Poisson systems with critical growth

“…The nonlinearity | | −2 ( > 3) is C 1 smooth. As regards other relevant papers for smooth nonlinearities about , we mention here [7,8,17,28]. Later, the differentiability of the nonlinearity was weaken by Alves et al in [1] and they dealt (SP) with K ( ) = 1 and ( ) = ( ) continuous and discussed the existence of ground states when V is periodic and asymptotically periodic in the meaning that there exists a periodic function V such that lim | |→∞ |V ( ) − V ( )| = 0.…”

Ground state solutions for asymptotically periodic Schrödinger–Poisson systems involving Hartree-type nonlinearities

“…When V (x) ≡ 1 and g(x, u) = |u| p-2 u, a radial positive solution was obtained for 4 < p < 6 in [9,10]. Later, Ruiz [11] proved the existence of a positive radial solution for 3 < p ≤ 4 by introducing the Nehari-Pohozaev manifold and establishing a key inequality.…”