Asymptotically periodic quasilinear Schrödinger equations with critical exponential growth

Abstract

In this work, we study the existence of a positive solution for a class of quasilinear Schrödinger equations involving a potential that behaves like a periodic function at infinity and the nonlinear term may exhibit critical exponential growth. In order to prove our main result, we combine minimax methods with a version of the Trudinger–Moser inequality. These equations appear naturally in mathematical physics and have been derived as models of several physical phenomena.

Existence of solutions to quasilinear Schrodinger equations with exponential nonlinearity

“…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.…”

“…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].…”

“…Problems involving (1.5) with critical nonlinearities in dimension N ≥ 3 have been addressed for instance in [10,17,18]. For dimension two, we can cite [23,24], which consider the nonlinearity with exponential critical growth. However, in these works the authors suppose that the nonlinearity is continuous.…”

“…Condition (A8) is inspired by the Trudinger-Moser inequality in the whole space (see Lemma 2.3). Note that the growth (A8) allows f (x, s) to behave as e (s 2 ) α at infinity, which is the exponential critical growth for this class of problems, for more details see [23,24].…”

“…The proof of the above lemma can be found in [22,23], so we omit it here. Using the change of variable v = G(u), a simple calculation shows that we can transform (1.1) into the nonhomogeneous semilinear equation…”