Multiplicity of positive solutions for a Kirchhoff type problem without asymptotic conditions - Qian, Xiaotao (Topological Methods in Nonlinear Analysis, 2023)
In this paper, we are concerned with the multiplicity of positive solutions for the following Kirchhoff type problem\[\begin{cases}-\bigg({\varepsilon}^2a+{\varepsilon}b\int_{\mathbb{R}^3} | u|^2dx\bigg)\Delta u+u=Q(x)|u|^{p-2}u, & x\in\mathbb{R}^3,\\u\in H^1\big(\mathbb{R}^3\big), \quad u> 0, & x\in\mathbb{R}^3,\end{cases}\]where $\varepsilon> 0$ is a small parameter, $a,b> 0$ are constants, $4< p< 6$, $Q$ is a nonnegative continuous potential and does not satisfy any asymptotic condition. Combining Nehari manifold and concentration compactness principle, we study how the shape of the graph of $Q(x)$ affects the number of positive solutions.