Existence of a positive solution to Kirchhoff problems involving the fractional Laplacian
Summary: The goal of this paper is to establish the existence of a positive solution to the following fractional Kirchhoff-type problem \[ \left(1+\lambda\int_{\mathbb{R}^N} \left( \big|(-\Delta)^{\frac{\alpha}{2}}u(x) \big|^{2}+V(x)u^2 \right)dx \right)\big[(-\Delta)^\alpha u+V(x)u\big]=f(u) \quad \text{in } \mathbb{R}^N, \] where \(N\geq 2\), \(\lambda\geq 0\) is a parameter, \(\alpha\in(0,1)\), \((-\Delta)^\alpha\) stands for the fractional Laplacian, \(f\in C(\mathbb{R}_+,\mathbb{R}_+)\). Using a variational method combined with suitable truncation techniques, we obtain the existence of at least one positive solution without compactness conditions.
scientific article; zbMATH DE number 6510072