A note on quasilinear Schrödinger equations with singular or vanishing radial potentials

Abstract

In this note we complete a previous study, where we got existence results for the quasilinear elliptic equation \begin{equation*} -Δw+ V\left( \left| x\right| \right) w - w \left( Δw^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 $r\to 0$, 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 $q_1=q_2$.

arXiv admin note: substantial text overlap with arXiv:2202.01872