Existence and Nonexistence of Solution for a Class of Quasilinear Schrödinger Equations with Critical Growth

Existence of solutions to quasilinear Schrodinger equations with exponential nonlinearity

“…In this article, under certain assumptions on g(s), V (x), f (x, s), and h(x), and by applying a fixed point theorem (see Lemma 2.6), as in [22] we show that (1.1) admits at least one weak solution. Here, the potential V (x) has similar characteristics as in [22]. The nonlinear term f (x, s) can be discontinuous and the critical exponential growth is allowed for it.…”

“…where w : R × R N → C is the unknown function, W : R N → R is a given potential, ρ : R + → R and p : R N × R + → R are real functions satisfying suitable conditions. Equation (1.3) has modeled many physical phenomena depending on the function ρ; for details see [3,21,22,25].…”

“…Motivated by these physical and mathematical aspects, Equation (1.5) has attracted the attention of numerous researchers, leading to results of existence and multiplicity of solutions. Noteworthy contributions include the works [10,15,22,25] in dimensions N ≥ 3 and [23,24] in the plane. In the later ones, the nonlinearity p(x, u) is continuous and exhibits exponential critical growth in the sense of Trudinger-Moser inequality.…”

“…Moreover, these studies assume that the potential V (x) is continuous, bounded from below and the main results are obtained by exploiting variational methods. In [22], the authors studied (1.1), with λ = 0 and in dimension N ≥ 3, considering potentials V (x) that can be discontinuous and singular. Furthermore, they allow the nonlinearity to be discontinuous and to have critical growth.…”

“…We emphasize that the context in dimension two is more delicate because it makes no sense to work in the space D 1,2 (R 2 ) and embedding available. Therefore, the strategy used to apply the fixed theorem is different from that of [22]. Our intention is to complement the study carried out in [22] and extend the results obtained in [23,24].…”