Singularly perturbed quasilinear Schrödinger equations with negative parameters

ABSTRACT

We consider a class of quasilinear Schrödinger equations of the form $$\begin{array}{c}\hfill -{\epsilon }^2\text{div}(g^2(u)\mathrm{\nabla }u)+{\epsilon }^2{g}^{\prime }(u)g(u){|\mathrm{\nabla }u|}^2+V(x)u=f(u),x\in {\mathbb{R}}^N,\end{array}$$

where $g^2(u)=1+\frac{\kappa u^2}{2(1+u^2)},\kappa \in (-2,0)$ and $N\ge 3.$ By variational methods, we construct a solution $u_{\epsilon }$ which concentrates, as $\epsilon \to 0,$ around the local minima of the potential V(x). One of our contributions is that we give the expression of some constant in our main result.

KEYWORDS:

• Schrödinger equations
• local mountain passes
• singular perturbations

AMS SUBJECT CLASSIFICATIONS:

• 35J20
• 35J60
• 35Q55

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

Supported by NSFC [grant number 11371146]; the Fundamental Research Funds for the Central Universities [grant number 2015zz133].