Supercritical Schrödinger equations with vanishing potential and double weighted nonlocal interaction
Year: 2025
Type: article
Abstract: In this paper, we discuss the existence of positive solutions for the following class of Schrödinger equations involving double weighted nonlocal interaction$\begin{align} -\Delta u+V(x)u = \left(\int_{\mathbb{R}^N}\frac{F(u)}{|y|^{\alpha}|x-y|^{\mu}}\,\mathrm{d}y\right)\frac{f(u)}{|x|^{\alpha}} + \psi(x,u), \quad \text{in } \mathbb{R}^{N}, \end{align}$where $ N\geq 3 $, $ 0<\mu <N $, $ \alpha\geq 0 $ and $ 0<2\alpha+\mu < \min\{\frac{N+2}{2},4\} $, the potential $ V:\mathbb{R}^N \rightarrow \mathbb{R} $ decays to zero at infinity and $ F $ is the primitive of function $ f. $ Under two types of supercritical function $ \psi $, we obtain the existence of solutions of the equation with subcritical or critical homogeneous $ f $ in the sense of the Stein-Weiss inequality. The approach is based on variational methods combined with a truncation argument, an adapted version of penalization technique and the Moser's iteration method.
DOI: 10.3934/cpaa.2025088
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4413370966