Explicit Formula Of The Critical Mass And The Energy Ground State Solution For The Mixed Local-Nonlocal Schrodinger Equation For The In-Between Critical Exponents Case

1. Problem Statement

The paper investigates the stationary solutions (standing waves) of the mixed local-nonlocal Schrödinger equation with a power nonlinearity $|u|^{p-2}u$. The equation is given by:
$$-\Delta u + (-\Delta)^s u + \omega u = |u|^{p-2}u, \quad x \in \mathbb{R}^N$$
where:

• $N \geq 3$ is the spatial dimension.
• $s \in (0, 1)$ is the fractional order.
• $(-\Delta)^s$ denotes the fractional Laplacian.
• The exponent $p$ lies in the "in-between critical" regime: $2 + \frac{4s}{N} < p < 2 + \frac{4}{N}$.

Context:

• $2 + \frac{4}{N}$ is the mass-critical exponent for the classical Laplacian ($-\Delta$).
• $2 + \frac{4s}{N}$ is the mass-critical exponent for the fractional Laplacian ($(-\Delta)^s$).
• In the mixed operator setting, the exponent $2 + \frac{4}{N}$ remains the mass-critical threshold. However, the presence of the fractional term introduces a new phenomenon in the interval $(2 + \frac{4s}{N}, 2 + \frac{4}{N})$.

Objective:
The authors aim to characterize the critical mass $c_0$ (the threshold mass below which no ground state exists) and the energy ground state solution $u_{c_0}$ (the minimizer of the energy functional at the critical mass). Specifically, they seek an explicit formula for $c_0$ and establish the relationship between the ground state and the optimizer of the associated Gagliardo-Nirenberg inequality.

2. Methodology

The authors employ a variational approach combined with sharp functional inequalities and scaling arguments.

• Variational Framework: The problem is formulated as minimizing the energy functional $J(u)$ on the $L^2$-sphere $S_c = \{u \in H^1(\mathbb{R}^N) : \|u\|_{L^2}^2 = c\}$:
  $$J(u) = \frac{1}{2}\|\nabla u\|_{L^2}^2 + \frac{1}{2}\|u\|_{D^{s,2}}^2 - \frac{1}{p}\|u\|_{L^p}^p$$
  where $\|u\|_{D^{s,2}}^2$ is the Gagliardo seminorm associated with the fractional Laplacian.

• Gagliardo-Nirenberg Inequality: A central tool is the derivation and optimization of a specific Gagliardo-Nirenberg inequality tailored to the mixed operator:
  $$\|u\|_{L^p}^p \leq C_{1,s,p} \|u\|_{D^{s,2}}^{\alpha} \|u\|_{D^{1,2}}^{\beta} \|u\|_{L^2}^{p-2}$$
  where the exponents $\alpha$ and $\beta$ depend on $N, s, p$.

• Weinstein Functional: To prove the existence of an optimizer for the inequality, the authors define a Weinstein functional $W_p(u)$ and utilize symmetric decreasing rearrangement and rescaling techniques. They overcome the difficulty of the numerator having three terms (unlike standard two-term cases) by establishing a non-vanishing property for the minimizing sequence.

• Scaling Analysis: The authors use precise scaling arguments ($u_t(x) = t^{N/2}u(tx)$) to analyze the behavior of the energy functional $J(u_t)$ and determine the conditions under which the infimum $m_c = \inf_{u \in S_c} J(u)$ is negative, zero, or positive.

• Pohozaev and Nehari Identities: These identities are used to derive relationships between the kinetic energy terms (local and nonlocal) and the potential energy term at the critical mass.

3. Key Contributions and Results

A. Existence of Critical Mass and Ground States (Theorem 1.1)

The paper rigorously proves that for the in-between critical exponent case ($2 + \frac{4s}{N} < p < 2 + \frac{4}{N}$):

• There exists a critical mass $c_0 > 0$.
• If $c < c_0$, no energy ground state solution exists ($m_c = 0$ is not attained, or $m_c = +\infty$ depending on the specific formulation, but effectively no minimizer).
• If $c \geq c_0$, an energy ground state solution exists.
• This establishes a sharp threshold phenomenon distinct from the pure fractional or pure local cases.

B. Explicit Formula for the Critical Mass (Theorem 1.3)

The primary contribution is the derivation of an explicit formula for the critical mass $c_0$ in terms of the best constant $C_{1,s,p}$ of the mixed Gagliardo-Nirenberg inequality:
$$c_0 = C_{1,s,p}^{-\frac{2}{p-2}} \left[ \frac{2N + 4 - pN}{2p(1-s)} \right]^{-\frac{2N+4-pN}{2(1-s)(p-2)}} \left[ \frac{pN - 2N - 4s}{2p(1-s)} \right]^{-\frac{pN-2N-4s}{2(1-s)(p-2)}}$$
This formula quantifies exactly how the critical mass depends on the dimension $N$, the fractional order $s$, and the nonlinearity power $p$.

C. Optimality and Ground State Equivalence

The authors prove that:

1. The optimizer $Q$ of the Gagliardo-Nirenberg inequality (after suitable scaling) is an energy ground state solution with mass $c_0$.
2. Conversely, the energy ground state solution $u_{c_0}$ with critical mass is an optimizer of the Gagliardo-Nirenberg inequality.
3. At the critical mass, the energy of the ground state is exactly zero ($J(u_{c_0}) = 0$).

D. Extension to Mixed Fractional Operators (Theorem 1.6)

The methodology is extended to the more general case of mixed fractional Laplacians $(-\Delta)^{s_1} + (-\Delta)^{s_2}$ (where $0 < s_1 < s_2 < 1$). The authors provide the explicit formula for the critical mass $\bar{c}_0$ and the corresponding best constant for this generalized setting, showing the robustness of their approach.

4. Significance

• Resolution of a New Phenomenon: The paper addresses a gap in the literature regarding the "in-between" critical regime where local and nonlocal diffusion terms compete. Previous works (e.g., Luo-Hajaiej) handled cases where exponents were below or above the fractional critical threshold, but the specific mixed operator case with $p$ between the two critical exponents was unresolved.
• Quantitative Precision: Unlike many existence proofs that rely on abstract variational arguments, this paper provides explicit, computable formulas for the critical mass. This is crucial for applications in physics and biology where specific parameter values determine system stability.
• Methodological Innovation: The proof of the existence of the optimizer for the three-term Gagliardo-Nirenberg inequality introduces novel arguments to handle the non-vanishing of minimizing sequences, overcoming technical hurdles present in standard two-term inequalities.
• Applications: The mixed local-nonlocal Schrödinger equation models phenomena in population dynamics (classical vs. non-local diffusion) and epidemiology (regional vs. global spread restrictions). The results provide a theoretical foundation for understanding the stability and existence of standing waves in these complex dispersal models.

In summary, this work provides a complete characterization of the ground state solutions for mixed local-nonlocal Schrödinger equations in the critical intermediate regime, offering explicit formulas for the critical mass and establishing a deep connection between variational minimizers and functional inequality optimizers.