[Paper Review] Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential

Helin Guo, Yimin Zhang+1 more|arXiv (Cornell University)|Jul 8, 2017

Advanced Mathematical Physics Problems15 references3 citations

TL;DR

This paper establishes the existence of normalized $L^2$-norm solutions for a nonlocal Kirchhoff-type elliptic equation with a trapping potential, proving that minimizers exist for all $\beta > 0$ when $p < 8/N$ and analyzing their blow-up behavior as $p \nearrow 8/N$. The key contribution is the explicit derivation of the critical threshold $\beta^*$ for existence across all $p \in (0, 8/N)$, revealing that the trapping potential fundamentally alters the solution structure compared to the $V \equiv 0$ case, where existence depends on $\beta$ being large enough.

ABSTRACT

We study a generalized Kirchhoff type equation with trapping potential. The existence and blow-up behavior of solutions with normalized L2-norm for this problem are discussed.

Motivation & Objective

• To extend the existence and asymptotic analysis of normalized $L^2$-norm solutions from the local case ($b=0$) to the nonlocal Kirchhoff-type equation ($b>0$) with a trapping potential $V(x) \geq 0$.
• To determine whether the presence of a non-zero trapping potential fundamentally changes the existence and blow-up behavior of solutions compared to the $V \equiv 0$ case.
• To derive an explicit expression for the critical threshold $\beta^*$ such that minimizers of the constrained energy functional exist if and only if $\beta > \beta^*$, for all $p \in (0, 8/N)$ and $1 \leq N \leq 4$.
• To analyze the blow-up profile of minimizers as $p \nearrow p^* = 8/N$, showing concentration at a point where $V(x) = 0$.

Proposed method

• Formulates the problem as a constrained minimization of the energy functional $E_p^\beta(u)$ over the set $S_1$ of functions with $\|u\|_{L^2} = 1$, using the constrained variational method.
• Applies Schwarz symmetrization to reduce the energy functional and establish radial symmetry of minimizers.
• Employs energy estimates and comparison with a limiting problem to analyze the blow-up behavior as $p \nearrow 8/N$, using rescaling $\overline{w}_p(x) = \epsilon_p^{N/2} u_p(\epsilon_p x + \epsilon_p \overline{y}_{\epsilon_p})$.
• Uses Fatou’s Lemma and the decay properties of $V(x)$ at infinity to prove that the concentration point $z_0$ must satisfy $V(z_0) = 0$, showing that blow-up occurs at a potential well minimum.
• Derives the asymptotic limit $\overline{w}_0$ of the rescaled minimizers, which solves a limiting equation involving $-b\Delta \overline{w}_0 = -\frac{b(4-N)}{2N}\overline{w}_0 + \beta^* \overline{w}_0^{p^*+1}$, and proves uniqueness up to translation.
• Establishes the existence of minimizers for all $\beta > 0$ when $V(x) \not\equiv 0$, contrasting sharply with the $V \equiv 0$ case where existence depends on $\beta > \beta^*$.

Experimental results

Research questions

• RQ1 Does the presence of a non-zero trapping potential $V(x)$ alter the existence threshold $\beta^*$ for normalized solutions in the nonlocal Kirchhoff equation compared to the $V \equiv 0$ case?
• RQ2 What is the precise asymptotic behavior of minimizers as $p \nearrow 8/N$, and where does the solution concentrate in the limit?
• RQ3 Can an explicit expression for $\beta^*$ be derived for all $p \in (0, 8/N)$, including the range $p \in (4/N, 8/N)$, where previous methods failed?
• RQ4 How does the blow-up profile of the solution relate to the geometry of the potential $V(x)$, particularly the location of its minimum?
• RQ5 Is the minimizer unique up to translation in the blow-up limit, and what equation does the limit profile satisfy?

Key findings

• The constrained minimization problem $d_\beta(p) = \inf_{u \in S_1} E_p^\beta(u)$ has a minimizer for all $\beta > 0$ and all $p \in (0, 8/N)$ when $V(x) \not\equiv 0$, in contrast to the $V \equiv 0$ case where existence depends on $\beta > \beta^*$.
• An explicit expression for the critical threshold $\beta^*$ is derived for all $p \in (0, 8/N)$, extending previous results limited to $p \in (0, 4/N]$, and valid for $1 \leq N \leq 4$.
• As $p \nearrow 8/N$, the minimizers $u_p$ concentrate at a point $z_0$ where $V(z_0) = 0$, and the rescaled profile $\overline{w}_p$ converges in $H^1(\mathbb{R}^N)$ to a unique positive solution $\overline{w}_0$ up to translation.
• The limit profile $\overline{w}_0$ satisfies the equation $-b\Delta \overline{w}_0 = -\frac{b(4-N)}{2N}\overline{w}_0 + \beta^* \overline{w}_0^{p^*+1}$, which is a modified nonlinear Schrödinger-type equation.
• The energy difference $d_\beta(p) - \widetilde{d}_\beta(p)$ vanishes as $p \nearrow p^*$, indicating that the energy of the minimizer approaches that of the limiting problem.
• The concentration point $\epsilon_p \overline{y}_{\epsilon_p}$ remains bounded and converges to a point $z_0$ with $V(z_0) = 0$, proving that blow-up occurs precisely at a potential minimum.