Solutions to a gauged Schrödinger equation with concave–convex nonlinearities without (AR) condition

Abstract

In this paper, we are concerned with a gauged nonlinear Schrödinger equation $\begin{array}{rl}& -\mathrm{\Delta }u+V\left(x\right)u+\lambda (\frac{h^2\left(|x|\right)}{|x{|}^2}+{\int }_{|x|}^{\mathrm{\infty }}\frac{h\left(s\right)}{s}{u}^2(s\left)\mathrm{d}s\right)\\ & u=\mu g(x,u)+\nu f(x,u),\mathrm{i}\mathrm{n}{\mathbb{R}}^2,\end{array}$ where $\lambda ,\mu ,\nu$ are positive parameters and $h\left(s\right)=\frac{1}{2}{\int }_0^{s}r{u}^2\left(r\right)\mathrm{d}r.$ Under appropriate assumptions on the potential V and nonlinear terms g and f, where g has sublinear growth and f has asymptotically linear or superlinear growth without (AR) condition, we acquire the existence and multiplicity of solutions by means of Mountain pass theorem and Fountain theorem. Our results generalize and improve the recent result in the literature.

Keywords:

• Gauged nonlinear Schrödinger equation
• concave–convex nonlinearities
• fountain theorem

2000 MSC Subject Classifications:

• 35J20
• 35J60

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This paper was supported financially by the Youth Science Foundation of China [grant number 11201272], Shanxi Province Science Foundation [grant number 2015011005] and Shanxi Scholarship Council of China [grant number 2016-009].