Bubble tower phenomena for an almost critical Hénon type problem

ABSTRACT

In this paper, we consider the following Hénon type problem $\left\{\begin{array}{rl}& -\mathrm{\Delta }u=|x{|}^{\alpha }{u}^{p_{\alpha }+\epsilon }+\lambda |x{|}^{\beta }u\mathrm{in}{B}_1\left(0\right),\\ & u>0,\mathrm{in}{B}_1\left(0\right),\\ & u=0,\mathrm{on}\mathrm{\partial }{B}_1\left(0\right),\end{array}\right.$ where $\alpha \ge 0$, $\epsilon >0$, $\lambda >0$, $-2<\beta \le N-4$, $p_{\alpha }=\left(N+2+2\alpha \right)/\left(N-2\right)$, $N\ge 3$, $B_1\left(0\right)$ is the unit ball of ${\mathbb{R}}^N$ centered at the origin. We show that if $\epsilon >0$ is small enough and λ goes to 0 at suitable rate, the above problem has radial bubble tower solutions, which behave like a superposition of bubbles and blow up at the origin.

COMMUNICATED BY:

• Daomin Cao

KEYWORDS:

• Bubble tower solutions
• Hénon type problem
• Lyapunov–Schmidt reduction

AMS 2000 SUBJECT CLASSIFICATIONS:

• Primary $35\mathrm{J}60$
• Secondary $35\mathrm{JB}05$
• $35\mathrm{J}40$

Acknowledgements

The author is very grateful to the referee for a number of valuable comments and suggestions, which improve the paper greatly.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by National Natural Science Foundation of China (NSFC) [grant number 11501166].