Finite Morse index solutions and asymptotics of weighted nonlinear elliptic equations. (English) Zbl 1275.35102

Summary: By introducing a suitable setting, we study the behavior of finite Morse-index solutions of the equation \[ -\operatorname{div}(| x|^\theta\nabla v)=| x|^l| v|^{p-1}v \quad \text{in }\Omega\subset\mathbb R^N \quad (N\geq 2),\tag{1} \] where \(p>1, \theta, l\in\mathbb{R}^1\) with \(N+\theta>2, l-\theta>-2\), and \(\Omega\) is a bounded or unbounded domain. Through a suitable transformation of the form \(v(x)=|x|^\sigma u(x)\), equation (1) can be rewritten as a nonlinear Schrödinger equation with Hardy potential \[ -\Delta u=| x|^\alpha | u|^{p-1}u+\frac{\ell}{| x|^2} u \quad \text{in }\Omega\subset\mathbb R^N \quad (N\geq 2),\tag{2} \] where \(p>1, \alpha \in (-\infty, \infty)\), and \(\ell \in (-\infty, (N-2)^2/4)\).
We show that under our chosen setting for the finite Morse-index theory of (1), the stability of a solution to (1) is unchanged under various natural transformations. This enables us to reveal two critical values of the exponent \(p\) in (1) that divide the behavior of finite Morse-index solutions of (1), which in turn yields two critical powers for (2) through the transformation. The latter appear difficult to obtain by working directly with (2).


35J60 Nonlinear elliptic equations
35B45 A priori estimates in context of PDEs
35J40 Boundary value problems for higher-order elliptic equations
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