We consider the critical Hénon equation \(\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u=|x|^\alpha |u|^{\frac{4}{N-2}}u\;\;& \text {in}\;B_1,\\ u=0\;\;& \text {on}\;\partial B_1, \end{array}\right. } \end{aligned}\) where \(\alpha >0\) , \(B_1\) is the unit ball centered at the origin in \(\mathbb {R}^N\) , \(N\ge 3\) . It is well-known that the above problem admits a unique positive radial solution \(u_\alpha \) . We first study the asymptotic behavior of \(u_\alpha \) as \(\alpha \rightarrow 0\) , which highlights a new interesting phenomenon of the Hénon equation and provides an affirmative answer to Hirano’s conjecture in 2009. Furthermore, we show that if \(\alpha \) is small enough, \(u_\alpha \) is non-degenerate and the Morse index of \(u_\alpha \) is \(N+1\) . These show the qualitative properties of \(u_\alpha \) are significantly different from those of solutions to power subcritical elliptic problem.
Based on above qualitative results, we can build infinitely many non-radial sign-changing bubbling solutions to the critical Hénon equation in \(\mathbb {R}^5\) , whose energy can be arbitrarily large. It seems that this is the first existence result of non-radial sign-changing solutions to the critical Hénon equation.
