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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).

MSC:

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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