[Paper Review] Finite Morse index solutions and asymptotics of weighted nonlinear elliptic equations

Yihong Du, Zongming Guo|arXiv (Cornell University)|Mar 18, 2013

Advanced Mathematical Physics ProblemsMathematics22 citations

TL;DR

This paper establishes a framework for analyzing finite Morse index solutions of weighted nonlinear elliptic equations via a transformation to a nonlinear Schrödinger equation with Hardy potential. It identifies two critical exponents for the power $p$ that divide the qualitative behavior of solutions, revealing stability and asymptotic properties that are difficult to obtain directly from the transformed equation.

ABSTRACT

By introducing a suitable setting, we study the behavior of finite Morse index solutions of the equation \[ -\{div} (|x|^θ abla v)=|x|^l |v|^{p-1}v \;\;\; \{in $Ω\subset \R^N \; (N \geq 2)$}, \leqno(1) \] where $p>1$, $θ, l\in\R^1$ with $N+θ>2$, $l-θ>-2$, and $Ω$ is a bounded or unbounded domain. Through a suitable transformation of the form $v(x)=|x|^σu(x)$, equation (1) can be rewritten as a nonlinear Schrödinger equation with Hardy potential $$-Δu=|x|^α|u|^{p-1}u+\frac{\ell}{|x|^2} u \;\; \{in $Ω\subset \R^N \;\; (N \geq 2)$}, \leqno{(2)}$$ where $p>1$, $α\in (-\infty, \infty)$ 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).

Motivation & Objective

• To develop a consistent finite Morse index theory for weighted elliptic equations with singular or degenerate coefficients.
• To analyze the asymptotic behavior and stability of solutions in punctured domains, exterior domains, and the full space.
• To identify critical values of the exponent $p$ that separate different solution behaviors, particularly regarding singularity removability and stability.
• To demonstrate that the transformation to the Schrödinger equation with Hardy potential preserves solution stability, enabling clearer identification of critical powers.
• To resolve limitations in prior work where critical exponents were difficult to derive directly from the Schrödinger form due to complex parameter dependence.

Proposed method

• Introduce a transformation $v(x) = |x|^\sigma u(x)$ to convert the weighted equation into a nonlinear Schrödinger equation with Hardy potential.
• Define finite Morse index in a setting that preserves stability under natural transformations, ensuring consistency across equivalent formulations.
• Use Kelvin transformation to relate solutions in exterior domains to those in punctured balls, enabling symmetry-based analysis.
• Apply comparison and test function techniques with carefully chosen cut-off functions to derive integral estimates and contradictions for non-removable singularities.
• Analyze the function $f_\tau(p)$ related to the spectral gap of the angular operator to determine stability thresholds.
• Establish critical exponents $\tilde{p}_c(N', \tau)$ and $p_c(N, 0)$ that separate stable, unstable, and singular solution regimes.

Experimental results

Research questions

• RQ1 What are the critical values of $p$ that determine the stability and asymptotic behavior of finite Morse index solutions to the weighted elliptic equation?
• RQ2 How does the transformation to the Schrödinger equation with Hardy potential affect the Morse index and stability of solutions?
• RQ3 Under what conditions is an isolated singularity at the origin removable for finite Morse index solutions?
• RQ4 Why are the critical exponents for the Schrödinger equation form difficult to derive directly, and how does the original weighted form simplify this?
• RQ5 What role does the Kelvin transformation play in extending results from bounded punctured domains to exterior domains?

Key findings

• Two critical exponents emerge: $\tilde{p}_c(N', \tau)$ and $p_c(N, 0)$, which partition the behavior of finite Morse index solutions in the weighted equation.
• For $p$ in the range $\tilde{p}_c(N', \tau) < p < \min\left\{\frac{N'+2+2\tau}{N'-2}, p_c(N, 0)\right\}$, finite Morse index solutions have removable singularities at the origin.
• When $p \in \left(\frac{N'+\tau}{N'-2}, \tilde{p}_c(N', \tau)\right]$, a stable positive solution with an isolated singularity at the origin exists.
• The transformation $v(x) = |x|^\sigma u(x)$ preserves solution stability, allowing the identification of critical powers that are otherwise obscured in the Schrödinger form.
• The function $f_\tau(p)$, related to the spectral gap, determines stability: $f_\tau(p) \leq \frac{(N'-2)^2}{4}$ implies stability.
• Contradiction arguments using test functions and logarithmic growth in $r$ confirm that solutions with non-removable singularities lead to divergent integrals, invalidating finite Morse index.