Distribution solutions of a static dispersion Schrödinger equation

Technical Summary: Distribution Solutions of a Static Dispersion Schrödinger Equation

Problem Statement
This paper investigates the qualitative properties of distribution solutions to a fourth-order static equation arising from the mixed dispersion Schrödinger equation, which models the propagation of intense laser beams in bulk media with Kerr nonlinearity. The primary focus is on the equation:
$$-\Delta u(x) + a^2 \Delta^2 u(x) = u^q(x), \quad u(x) > 0 \text{ in } \mathbb{R}^3,$$
where $a > 0$ and $q > 0$. This equation serves as the Euler-Lagrange equation for extremal functions of a specific embedding inequality involving the space $D = \{u \in D^{1,2}(\mathbb{R}^3) : \Delta u \in L^2(\mathbb{R}^3)\}$. The authors also examine an Allen-Cahn-type variant: $( -\Delta + a^2 \Delta^2 )u = u(1-u^2)$.

The central challenges addressed include:

1. Establishing the equivalence between the partial differential equation (PDE) and an associated integral equation involving a modified Coulomb potential kernel.
2. Determining the existence and non-existence (Liouville theorems) of positive solutions based on the exponent $q$.
3. Analyzing the regularity (differentiability, integrability, radial symmetry) and asymptotic decay rates of these solutions at infinity.
4. Investigating the attainability of the best constant in the associated embedding inequality.

Methodology
The authors employ a combination of elliptic regularity theory, integral equation analysis, and variational methods:

• Equivalence of PDE and Integral Equation: The study establishes that the PDE is equivalent to the integral equation $u(x) = \int_{\mathbb{R}^3} K_a(x-y)u^q(y)dy$, where $K_a(x) = \frac{1-e^{-|x|/a}}{|x|}$ is the kernel solving $-\Delta \phi + a^2 \Delta^2 \phi = 4\pi \delta_0$. This equivalence relies on the properties of the kernel and density arguments in the space $D$.
• Liouville Theorems via Pohozaev Identity: To derive non-existence results for certain ranges of $q$, the authors utilize the Pohozaev identity in integral form. This approach avoids the complex calculations associated with third-order derivatives in the standard PDE Pohozaev identity, particularly for the Allen-Cahn-type equation.
• Regularity Lifting and Moving Planes: The regularity of solutions (integrability and differentiability) is proven using a regularity lifting lemma and the method of moving planes in integral form. These techniques allow the authors to establish that solutions belong to $L^s(\mathbb{R}^3)$ for various $s$ and possess radial symmetry.
• Asymptotic Analysis: The decay rates at infinity are estimated by comparing the solution $u(x)$ with the kernel $K_a(x)$, utilizing the specific properties of the modified Coulomb potential.

Key Contributions and Results

1. Equivalence Result (Theorem 1.1): The paper proves that a positive distribution solution $u \in D$ of the PDE satisfies the integral equation almost everywhere if and only if $u \in L^{3(q-1)/2}(\mathbb{R}^3)$. Conversely, a positive solution of the integral equation in this space is a distribution solution of the PDE.

2. Liouville Theorems and Critical Exponents (Theorems 1.2 & 1.3):

   • For the PDE, if a positive distribution super-solution exists in $D$, then $q > 3$. If a positive distribution solution exists, then $q > 5$.
   • For the integral equation, a positive super-solution in $L^\infty_{loc}$ exists if and only if $q > 3$. A positive differentiable solution in $L^{q+1}$ exists only if $q > 5$.
   • These results identify $q=3$ as the Serrin exponent and $q=5$ as the Sobolev exponent for this problem.

3. Non-Attainability of Best Constants (Remark 1.2): A significant consequence of the Liouville theorems is that the best constant for the embedding inequality $\|u\|_{L^{q+1}} \leq C(\|\nabla u\|_{L^2} + a\|\Delta u\|_{L^2})$ is not attainable when $q=5$. This contrasts with the standard Hardy-Littlewood-Sobolev inequality where extremal functions exist. The lack of attainability is attributed to the absence of rescaling invariance in the modified Coulomb potential kernel.

4. Regularity and Symmetry (Theorem 1.5): For positive solutions of the integral equation with $q > 3$:

   • The solution belongs to $L^s(\mathbb{R}^3)$ for any $s \in (3, \infty]$, with 3 being the optimal lower bound.
   • The solution is radially symmetric and monotonically decreasing about some point $x_0$.
   • The solution decays at infinity at the same rate as the kernel $K_a(x)$, i.e., $C^{-1}K_a(x) \leq u(x) \leq C K_a(x)$ for large $|x|$.

5. Allen-Cahn Type Equation (Theorems 1.6 & 1.7): For the equation $( -\Delta + a^2 \Delta^2 )u = u(1-u^2)$, the authors prove that bounded uniformly continuous solutions must be trivial ($u \equiv 0$ or $u \equiv 1$) under specific integrability conditions on the nonlinearity. This is achieved using the integral form of the Pohozaev identity to avoid structural constraints required for the PDE version.

Significance and Claims
The paper claims to provide a comprehensive qualitative analysis of the static mixed dispersion Schrödinger equation, specifically addressing the gap in understanding the role of the fourth-order dispersion term combined with the Laplacian. The authors highlight that the operator $-\Delta$ plays a more prominent role than $a^2 \Delta^2$ in determining the existence of solutions, as evidenced by the critical exponents matching those of the second-order Lane-Emden equation rather than the pure fourth-order case.

The work is significant for mathematical physics as it clarifies the stability conditions for standing wave solutions and the scattering behavior of the evolution equation. Furthermore, the demonstration that the best constant of the associated embedding inequality is unattainable for the critical case $q=5$ offers a distinct contrast to classical Sobolev-type inequalities, attributing this phenomenon to the specific nature of the modified Coulomb potential. The methodology, particularly the application of the Pohozaev identity in integral form to handle higher-order and Allen-Cahn-type equations, is presented as a robust tool for avoiding tedious derivative calculations while maintaining rigorous structural conditions.