Positive solutions for generalized quasilinear Schrödinger equations with asymptotically linear nonlinearities

ABSTRACT

In this paper, we study the following generalized quasilinear Schrödinger equations $-\mathrm{d}\mathrm{i}\mathrm{v}\left(g^2\right(u\left)\mathrm{\nabla }u\right)+g\left(u\right)g^{\prime }\left(u\right)|\mathrm{\nabla }u{|}^2+V\left(x\right)u=h\left(u\right),x\in {\mathbb{R}}^N,$ where $N\ge 3$, $V\in C({\mathbb{R}}^{\mathbb{N}},{\mathbb{R}}^{\mathbb{+}})$ is a given potential, g is a $C^1$ even function with $g^{\prime }\left(t\right)\le 0$ for all $t\ge 0,g\left(0\right)=1,\underset{t\to +\mathrm{\infty }}{lim}g\left(t\right)=a,0<a<1$.The nonlinearity $h\in C^1(\mathbb{R},\mathbb{R})$ satisfies asymptotically linear at infinity. Under certain assumptions on V,g and h, we give the existence results via variational methods. Moreover, we give an application for our results.

KEYWORDS:

• Generalized Quasilinear Schrödinger equations
• asymptotically linear
• positive solutions

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35J20
• 35J62

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by National Natural Science Foundation of China [grant numbers 11471235, 11626164 and 11771319]. G. Li was partly supported by the Applied Basic Research Projects of Yunnan Local Colleges of China [grant number 2017FH001-011], the Foundation of Education of Commission of Yunnan Province of China [grant number 2018JS451], the Graduate Innovation Project of Jiangsu Province of China [grant number KYLX$16\mathrm{\_}0093$].