Abstract:

This paper study the following supercritical Hénon-type problem

where $B_1(0)$ is the unit ball in $\mathbb{R}^N$, $N \geq 5$, $\alpha > 0$, $p_{\alpha} = \frac{N+2+2\alpha}{N-2}$, and $\lambda \to 0$ as $\varepsilon \to 0$. By using the Emden-Fowler transformation and the Lyapunov-Schmidt reduction method, we construct bubble-tower solutions to this problem which are highly concentrated at the origin. we overcome the difficulty brought by the lack of compactness due to the supercritical growth of the nonlinearity. The Emden-Fowler transformation plays an important role in this work. it is not only a change of variables, but also a structural reconstruction in the sense of geometric analysis. This transformation converts the complicated bubble-tower concentration phenomenon near the origin into one have multi-peak solutions concentrated at separable points at infinity, so that the transformed equation is suitable for applying the Lyapunov-Schmidt reduction method, and thus we can construct solutions to the problem.

Key words: Lyapunov-Schmidt reduction, Emden-Fowler transformation, bubble-tower solutions