Asymptotic profiles for a nonlinear Kirchhoff equation with combined powers nonlinearity

We study asymptotic behavior of positive ground state solutions of the nonlinear Kirchhoff equation $$-\Big(a+b\int_{\mathbb R^N}|\nabla u|^2\Big)\Delta u+ \lambda u= u^{q-1}+ u^{p-1} \quad {\rm in} \ \mathbb R^N,$$ as $\lambda\to 0$ and $\lambda\to +\infty$, where $N=3$ or $N= 4$, $2<q\le p\le 2^*$, $2^*=\frac{2N}{N-2}$ is the Sobolev critical exponent, $a>0$, $b\ge 0$ are constants and $\lambda>0$ is a parameter. In particular, we prove that in the case $2<q<p=2^*$, as $\lambda\to 0$, after a suitable rescaling the ground state solutions of the problem converge to the unique positive solution of the equation $-\Delta u+u=u^{q-1}$ and as $\lambda\to +\infty$, after another rescaling the ground state solutions of the problem converge to a particular solution of the critical Emden-Fowler equation $-\Delta u=u^{2^*-1}$. We establish a sharp asymptotic characterisation of such rescalings, which depends in a non-trivial way on the space dimension $N=3$ and $N= 4$. We also discuss a connection of our results with a mass constrained problem associated to the Kirchhoff equation with the mass normalization constraint $\int_{\mathbb R^N}|u|^2=c^2$.