Existence of solution for a generalized quasilinear elliptic problem

Abstract

It establishes existence and multiplicity of solutions to the elliptic quasilinear Schrödinger equation −div(g2(u)∇u)+g(u)g′(u)|∇u|2+V(x)u=h(x,u),x∈ℝN,where g, h, V are suitable smooth functions. The function g is asymptotically linear at infinity and, for each fixed x∈ℝN, the function h(x, s) behaves like s at the origin and s3 at infinity. In the proofs, we apply variational methods.

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.…”

“…The existence of solutiona for equations of the form (1.5) has been discussed under various conditions on the potential V (x) and the nonlinear term p(x, s), see for instance [10,15,25,26]. It is usually assumed that the potential is continuous and another condition that guarantees some compactness result.…”

Positive ground state solutions for generalized quasilinear Schrödinger equations with critical growth

“…They establish the existence of positive solutions for (1.2) with subcritical nonlinear terms. Later, the results are extended by [10] for the existence and multiplicity of solutions, and by [6] for the existence of ground state solutions. As for generalized quasilinear Schrödinger equations with critical growth, Deng et al in [8] establish the existence of positive solutions by using the mountain pass theorems.…”