Positive solutions of Kirchhoff-Hénon type elliptic equations with critical Sobolev growth

Abstract

We investigate the following Kirchhoff-Hénon type equation involving the critical Sobolev exponent with Dirichlet boundary condition: \[ - \bigg( a + b \bigg( \int_\Omega \lvert Du \rvert^2 dx \bigg)^{(p-2)/2} \bigg) \Delta u = \Psi u^{q-1} + \lvert x \rvert^\alpha u^{2^* - 1} \] in $\Omega$ included in a unit ball under several conditions. Here, $a, b \geq 0$, $a + b > 0$, $2 < p < q < 2^*$ and $\Psi \in L^\infty(\Omega) \setminus \{ 0 \}$ is a given non-negative function with several conditions. We show that, if either $N = 3$ with $4 < q < 2^* = 6$ or $N \geq 4$, there exists a positive solution for small $\alpha \geq 0$. Our methods includes the mountain pass theorem and the Talenti function.