Existence of solution for an asymptotically linear Schrödinger-Kirchhoff equation (Q1740584)

The authors deal with the Kirchhoff equation \[ -\left(1+\lambda \int_{\mathbb{R}^N}|\nabla u|^2\,dx\right)\Delta u + V(x)u=f(u),\ x\in \mathbb{R}^N,\ N\in \{3,4\},\ \lambda\ge 0, \] where the potential \(V\) is radial and \(f\) can be superlinear or asymptotically linear at infinity. The main result is the theorem on the existence and the asymptotic behavior of a solution. \par In the case \(N=4\) there exists \(\lambda^* > 0\) such that for any \(\lambda \in (0,\lambda^*)\) the equation has a ground state solution \(u_\lambda \in H^1_{\mathrm{rad}}(\mathbb{R}^4)\). Moreover \(u_\lambda \to u_0\) in \(H^1_{\mathrm{rad}}(\mathbb{R}^4)\) as \(\lambda \to 0^+\), where \(u_0\) is a weak solution of \(\Delta u_0+V(x)u_0=f(u_0)\), \( x\in \mathbb{R}^4\). \par The same result holds for \(N=3\) with no restriction on \(\lambda\).