The best constant in the G-N inequality for the mixed local and Nonlocal Laplacian

1. Problem Statement

The paper addresses the determination of the best constant ($C_{N,p,s}$) for the Gagliardo-Nirenberg inequality involving a mixed local and nonlocal operator. Specifically, the authors study the inequality for functions $u$ in a mixed Sobolev space $E$:

$$\int_{\mathbb{R}^N} |u|^p dx \leq C_{N,p,s} \|u\|_{D^{s,2}(\mathbb{R}^N)}^{\frac{2N-p(N-2)}{2(1-s)}} \|u\|_{D^{1,2}(\mathbb{R}^N)}^{\frac{p(N-2s)-2N}{2(1-s)}}$$

where:

• $N \geq 3$, $s \in (0, 1)$, and $p \in (2^*_s, 2^*)$.
• $2^* = \frac{2N}{N-2}$ is the critical Sobolev exponent.
• $2^*_s = \frac{2N}{N-2s}$ is the fractional critical Sobolev exponent.
• The norms correspond to the homogeneous fractional Sobolev space $D^{s,2}$ (associated with the fractional Laplacian $(-\Delta)^s$) and the standard Sobolev space $D^{1,2}$ (associated with the classical Laplacian $\Delta$).

The underlying equation associated with the Euler-Lagrange equation of this inequality is:
$$-\Delta u + (-\Delta)^s u = |u|^{p-2}u, \quad x \in \mathbb{R}^N. \quad (P)$$

2. Methodology and Challenges

The authors employ variational methods and geometric analysis, but face significant hurdles distinct from previous works on pure local or pure nonlocal operators.

Key Challenges:

1. Regularity and Pohozaev Identity: In standard problems (e.g., pure Laplacian or pure fractional Laplacian), the existence of the $L^2$ term (mass) in the equation usually aids in proving regularity, which allows weak solutions to satisfy the Pohozaev identity. In equation (P), the $L^2$ term is absent. Consequently, it is difficult to establish the regularity of weak solutions, making it unclear if all weak solutions satisfy the Pohozaev identity.
2. Uniqueness and Non-degeneracy: The uniqueness of ground state solutions (modulo symmetries) and the non-degeneracy of the linearized operator for mixed operators are open problems. Standard proofs relying on the uniqueness of ground states (like Weinstein's method for the pure Laplacian) cannot be directly applied.
3. Linking Optimizers to Ground States: Establishing a direct relationship between the optimizer of the inequality and the ground state solution of equation (P) is non-trivial due to the lack of the aforementioned regularity and uniqueness results.

Innovative Approach:
To overcome these difficulties, the authors utilize the Nehari-Pohozaev manifold method.

• They define the Nehari manifold $\mathcal{N}$ (where the derivative of the energy functional vanishes in the direction of the function).
• They define the Nehari-Pohozaev manifold $\mathcal{M}$ (where the Pohozaev identity holds).
• Instead of assuming all weak solutions satisfy the Pohozaev identity, they prove the existence of a specific solution on $\mathcal{M}$ that minimizes the energy.
• They extend Lieb's translation theorem to the mixed Sobolev space $E$ to handle the lack of compactness in $\mathbb{R}^N$.

3. Key Contributions and Results

The paper establishes four main theorems:

Theorem 1.1: Existence of the Best Constant and Optimizer

• Proves the existence of a positive best constant $C_{N,p,s}$.
• Establishes the existence of a radial non-negative optimizer $u \in E$.
• Shows that a scaled version of this optimizer, $Q = \lambda_1 u(\lambda_2 x)$, is a weak solution to equation (P).
• Derives explicit relations between the norms of the optimizer and the best constant.

Theorem 1.2: Existence of Ground State Solutions

• Proves that equation (P) possesses a radial ground state solution $\Phi \in E$.
• This solution is found by minimizing the energy functional $I(u)$ on the Nehari manifold $\mathcal{N}$.
• The energy level is $c = \inf_{u \in \mathcal{N}} I(u)$.

Theorem 1.3: Existence of Solutions Satisfying Pohozaev Identity

• Proves the existence of a solution $\Psi \in E$ that satisfies the Pohozaev identity ($P(\Psi)=0$).
• This solution is found by minimizing the energy on the Nehari-Pohozaev manifold $\mathcal{M}$.
• Crucially, this result bridges the gap: it confirms that the ground state solution of the mixed operator equation does satisfy the Pohozaev identity, despite the lack of general regularity results.

Theorem 1.4: Explicit Formula for the Best Constant

• Combines the previous results to provide an explicit formula for the best constant $C_{N,p,s}$ in terms of the ground state energy $c$:
  $$C_{N,p,s}^{-1} = \left( \frac{2N - p(N-2)}{2p(1-s)} \right)^{\frac{2N-p(N-2)}{4(1-s)}} \left( \frac{p(N-2s) - 2N}{2p(1-s)} \right)^{\frac{p(N-2s)-2N}{4(1-s)}} \left[ \frac{2p}{p-2} c \right]^{\frac{p-2}{2}}$$
• It confirms that the optimizer of the inequality is not just a weak solution, but specifically a ground state solution that satisfies the Pohozaev identity.

4. Significance

• Resolution of Regularity Issues: The paper successfully bypasses the need for full regularity theory to prove the Pohozaev identity for the specific ground state solution. By constructing the solution on the Nehari-Pohozaev manifold, they ensure the identity holds by definition of the minimization path.
• Extension of Weinstein's Method: It extends the classical method of Weinstein (originally for the Laplacian) and the fractional Laplacian to the more complex mixed operator setting.
• Application to Physical Models: The mixed operator $-\Delta + (-\Delta)^s$ models phenomena involving both classical diffusion (local) and anomalous diffusion (nonlocal), such as population dynamics, epidemic spreading, and elastic body behavior. Knowing the sharp constant is vital for stability analysis and blow-up criteria in these physical models.
• Foundational for Future Work: The establishment of the existence and properties of ground states for mixed operators provides a necessary foundation for studying more complex problems, such as the stability of solitons or the behavior of solutions in bounded domains.

In summary, this paper provides a rigorous variational framework to determine the sharp Gagliardo-Nirenberg constant for mixed operators, overcoming the technical barrier of missing $L^2$ terms and lack of regularity by utilizing the Nehari-Pohozaev manifold approach.