Existence and multiplicity results for the nonlinear Schrödinger–Maxwell systems

Abstract

In this paper, we study the nonlinear Schrödinger-Maxwell systemwhere the potential V and the primitive of g are allowed to be sign-changing, and g is local superlinear. Under some simple assumptions on V, Q and g, we establish some existence criteria to guarantee that the aforementioned system has at least one nontrivial solution or infinitely many nontrivial solutions by using critical point theory. Recent results in the literature are generalized and significantly improved.Such a system is called Schrödinger-Maxwell equations or Schrödinger-Poisson equations, which is obtained while looking for the existence of standing waves for the nonlinear Schrödinger equations interacting with an unknown electrostatic field. System (1.1) arises in many mathematical physics contexts, such as in quantum electrodynamics, to describe the interaction of a charge particle with an electromagnetic field, and also in semiconductor theory, in nonlinear optics and in plasma physics. We refer to [1,2] and the references therein for more details on the physical aspects. If Q Á 0, System (1.1) becomes the nonlinear Schrödinger equation, which has been extensively studied in many recent papers via variational methods under the various hypotheses, see [3][4][5][6][7][8][9][10][11] and references therein. In recent years, System (1.1) (with Q 6 Á 0) has been widely investigated by using variational methods under various conditions on V, Q and g, see [1,[12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31] and references therein. The case V Á Q Á 1, and g.x, u/ D juj q 1 u, 1 < q < 5, has been investigated in [1,12,13,15,16,19,20,23,24]. More precisely, Ambrosetti [12] obtained the existence of infinitely many pairs of high-energy radial solutions when 2 < q < 5 and also obtained some existence results for the case when 1 < q < 2 and q D 2. D'Avenia [15] considered the existence of nonradial solutions. And in [30], the authors dealt with the critical exponent case.When V.x/ or Q.x/ is not a constant, Lv [22] and Sun [26] considered the case that g.x, u/ is sublinear at infinity in u and the positive potential V.x/ is nonradial. In [27], the authors considered the case that g.x, u/ is asymptotically linear at infinity in u and the positive potential V.x/ is bounded and nonradial. Moreover, infinitely many high-energy solutions for the superlinear case are obtained in [18,21,28] via the (variant) Fountain Theorem. More precisely, Li, Su and Wei [21] obtained the following theorem by using the variant Fountain Theorem. Assume that the following conditions are satisfied.(V 0 ) V 2 C.R 3 , R/ and inf x2R 3 V.x/ a 0 0 > 0, where a 0 0 is a positive constant. (V 1 ) For any T 0 > 0, meas.fx 2 R 3 : V.x/ Ä T 0 g/ < 1, where meas. / denotes the Lebesgue measure in R 3 .