A note on quasilinear Schr\"odinger equations with singular or vanishing radial potentials

TL;DR

This paper extends previous results on quasilinear Schrödinger equations with radial potentials by removing the assumption that the potential vanishes at the origin, using a change of variables and variational methods to establish existence of solutions.

Contribution

It removes the restriction on the potential's behavior at zero and proves existence of solutions for a broader class of radial potentials in quasilinear Schrödinger equations.

Findings

Existence of non-negative solutions in ^N  with singular or vanishing potentials.

Solutions are classical in ^N \u00d7 {0}.

Applicable to nonlinearities with double-power behavior.

Abstract

In this note we complete a previous study, where we got existence results for the quasilinear elliptic equation \begin{equation*} -\Delta w+ V\left( \left| x\right| \right) w - w \left( \Delta w^2 \right)= K(|x|) g(w) \quad \text{in }\mathbb{R}^{N}, \end{equation*} with singular or vanishing continuous radial potentials $V(r)$, $K(r)$. In our previuos study we assumed, for technical reasons, that $K(r)$ was vanishing as $r \rightarrow 0$, while in the present paper we remove this obstruction. To face the problem we apply a suitable change of variables $w=f(u)$ and we find existence of non negative solutions by the application of variational methods. Our solutions satisfy a weak formulations of the above equation, but they are in fact classical solutions in $\mathbb{R}^{N} \setminus \{0\}$. The nonlinearity $g$ has a double-power behavior, whose standard example is $g(t) = \min \{ t^{q_1 -1}, t^{q_2 -1} \}$ ($t>0$), recovering the usual case of a single-power behavior when $q_1 = q_2$.