Positive solutions to multi-critical Schr\"{o}dinger equations

TL;DR

This paper proves the existence of multiple positive solutions to a multi-critical Schrödinger equation with nonlinear and nonlocal terms, especially when a parameter is large, using topological methods.

Contribution

It establishes the existence of multiple positive solutions for multi-critical Schrödinger equations with nonlocal terms, expanding the understanding of solution multiplicity in such problems.

Findings

At least as many solutions as the category of the domain.

Solutions exist for large values of the parameter λ.

The problem involves critical exponents and nonlocal convolution terms.

Abstract

In this paper, we investigate the existence of multiple positive solutions to the following multi-critical Schr\"{o}dinger equation \begin{equation} \label{p} \begin{cases} -\Delta u+\lambda V(x)u=\mu |u|^{p-2}u+\sum\limits_{i=1}^{k}(|x|^{-(N-\alpha_i)}* |u|^{2^*_i})|u|^{2^*_i-2}u\quad \text{in}\ \mathbb{R}^N,\\ \qquad\qquad\qquad u\,\in H^1(\mathbb{R}^N), \end{cases} \end{equation} where $\lambda,\mu\in \mathbb{R}^+, \, N\geqslant 4$, and $2^*_i=\frac{N+\alpha_i}{N-2}$ with $N-4<\alpha_i<N,\,i=1,2,\cdots,k$ are critical exponents and $2<p<2^*_{min}=\min\{2^*_i:i=1,2,\cdots,k\}$. Suppose that $\Omega=int\,V^{-1}(0)\subset\mathbb{R}^N$ is a bounded domain, we show that for $\lambda$ large, problem above possesses at least $cat_\Omega(\Omega)$ positive solutions.