Perturbation limiting behaviors of ground states to the Kirchhoff equation with combined power-type nonlinearities

Abstract

In this paper, we consider the Kirchhoff-type equation with combined power-type nonlinearities:$\begin{array}{c}-(a+b\underset{R^N}{\int }|\nabla u{|}^{2}dx)\Delta u=\lambda u+|u{|}^{p-1}u-\epsilon |u{|}^{q-1}u\text{in }{R}^N,\end{array}$ where $a>0$, $b>0$, $\epsilon >0$ are constants, $\lambda \in R$, $1\le N\le 3$ and $2<p<q<2^{⁎}$. We mainly focus on the existence and perturbation limit behaviors of ground states $u_{\epsilon ,\rho }$, where $u_{\epsilon ,\rho }$ is radially symmetric-decreasing and ${\int }_{R^N}|u_{\epsilon ,\rho }{|}^{2}dx=\rho$. Firstly, we prove the existence and nonexistence of ground states by using the concentration-compactness principle. Secondly, we characterize the perturbation limit behaviors of ground states $u_{\epsilon ,\rho }$ as $\epsilon \to 0^{+}$ and find that the blow-up phenomenon happens for $2+8/N\le p<q<2^{⁎}$ and $\rho >{\rho }_c$ in the sense that ${\lim }_{\epsilon \to 0^{+}}{\int }_{R^N}|\nabla u_{\epsilon ,\rho }{|}^{2}dx\to +\infty$. Moreover, we obtain two different blow-up profiles corresponding to two limit equations. In particular, for mass super-critical $(2+8/N<p<q<2^{⁎})$, the limit profiles is given by the nonlinear Schrödinger Thomas-Fermi minimizer $\sqrt{{\rho }_{⁎}{1}_{\Omega }}\left(x\right)$, where $1_{\Omega }\left(x\right)$ denotes the characteristic function for the Borel set $\Omega =B(0,\sqrt[N]{\frac{N\rho }{{\rho }_{⁎}{\omega }_N}})$ and ${\rho }_{⁎}={\left(\frac{q(p-2)}{p(q-2)}\right)}^{2/(q-p)}$. Finally, we obtain the sharp blow-up rate for $u_{\epsilon ,\rho }$ that ${\Vert u_{\epsilon ,\rho }\Vert }_{L^{\infty }}\sim {\epsilon }^{-1/(q-p)}$ as $\epsilon \to 0^{+}$ with $2+8/N\le p<q<2^{⁎}$ and $\rho >{\rho }_c$.