Ground states for quasilinear Schrödinger equations with critical growth

Positive solutions of quasilinear Schrödinger equations with critical growth

“…By the same change of variables as in [5], critical quasilinear equations like (1.3) with periodic potential were studied by Silva and Vieira in [10]. Under the assumption p > max{ N+6 N−2 , 3}, Liu, Liu and Wang [11] established a existence result of solutions of (1.3) with a bounded potential via the Nehari method. In this paper, to obtain the standing wave solutions of (1.1), we look for solutions of (1.3).…”

“…In [5,11], the restriction 3 < p < 22 * − 1 was used essentially for the verification of boundedness of a Cerami sequence and to prove that the mountain pass level value is less than the threshold for compactness of a Cerami sequence.…”

Solutions for quasilinear Schrödinger systems with critical exponents

“…Due to its significant applications in mathematical physics (see [1][2][3] and the references therein), this type of quasilinear equations have been widely studied in the literature and lots of results are achieved, see for example, [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] and the references therein. On the other hand, the quasilinear Schrödinger system like (1.1) is also a challenging physical problem that has been studied in recent years, which describes the interaction between two electrons and includes the MNLS.…”

“…More precisely, there is no natural space in which we can define the variational functional of the problem such that it possesses both smoothness and compactness. Those specific mathematical difficulties attract many researchers, and several new ideas are used, such as minimization method [20,18], change of variables [21] and Nehari method [17]. However, the changing variable argument proves to be quite effective for dealing with the special problem as −Δu − Δ(u 2 )u = g (x, u) where the quasilinear term is exactly −Δ(u 2 )u, so they can be transformed to a semilinear problem and various techniques for semilinear problems can be adapted in this situation.…”

“…However, the changing variable argument proves to be quite effective for dealing with the special problem as −Δu − Δ(u 2 )u = g (x, u) where the quasilinear term is exactly −Δ(u 2 )u, so they can be transformed to a semilinear problem and various techniques for semilinear problems can be adapted in this situation. When we consider the Nehari method used in [17], we find that it requires certain monotonicity conditions of the equation. In this paper, we present a generalized perturbation approach which is originally due to [16] where the single equation was considered.…”