Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation (Q2309162)

The paper deals with the Dirichlet problem for the generalized Hénon equation \(-\Delta u = |x|^\alpha |u|^{p-2}u\) on the unit ball in \(\mathbb{R}^N\). Here \(p>2\) is fixed and \(\alpha>0\) is considered as parameter. In the subcritical range \(\alpha>\frac{(N-2)p-2N}{2}\) it has been proved in [\textit{K. Nagasaki}, J. Fac. Sci., Univ. Tokyo, Sect. I A 36, No. 2, 211--232 (1989; Zbl 0699.35097)] that for each \(K\in\mathbb{N}\) there exists a unique classical radial solution \(u_\alpha\) with precisely \(K\) nodal domains and \(u_\alpha(0)>0\). The main results of the paper are asymptotic \(C^1\)-expansions for the \(i\)-th negative eigenvalue \(\mu_i(\alpha)\) of the linearization of the equation at \(u_\alpha\) as \(\alpha\to\infty\). A consequence is the existence of an unbounded sequence of bifurcation points on the branch \((\alpha,u_\alpha)\). The bifurcating solutions are non-radial and have \(K\) nodal domains \(\Omega_1,\dots,\Omega_k\), where \(\Omega_1\) contains \(0\) and is homeomorphic to a ball, and \(\Omega_2,\dots,\Omega_K\) are homeomorphic to annuli. These results complement the bifurcation results from [\textit{A. L. Amadori} and \textit{F. Gladiali}, Adv. Differ. Equ. 19, No. 7--8, 755--782 (2014; Zbl 1320.35056)] where \(p\) is taken as bifurcation parameter.