The isocritical regime for mixed local-nonlocal $(p,q)$ Laplacian: existence of ground state, and decay estimates

Technical Summary: The Isocritical Regime for Mixed Local-Nonlocal $(p, q)$-Laplacian

Problem Statement
This paper investigates the existence and qualitative properties of solutions to the critical equation driven by the mixed local-nonlocal operator $L_{p,q} := -\Delta_p + (-\Delta)^s_q$ in the entire space $\mathbb{R}^N$:
$$-\Delta_p u + (-\Delta)^s_q u = |u|^{p^*-2}u \quad \text{in } \mathbb{R}^N,$$
where $N \ge 2$, $1 < p < N$, $0 < s < 1$, and $1 < sq < N$. The central focus is the isocritical regime, defined by the condition $p^* = q^*_s$, where $p^* = \frac{pN}{N-p}$ is the critical Sobolev exponent for the $p$-Laplacian and $q^*_s = \frac{qN}{N-sq}$ is the fractional critical exponent for the $q$-Laplacian. This condition is equivalent to $1 - \frac{N}{p} = s - \frac{N}{q}$.

In non-isocritical regimes (where $p^* \neq q^*_s$), one operator typically dominates the critical scaling, rendering the other a lower-order perturbation. The authors note that in previous work on the linear case ($p=q=2$), the nonlocal term was shown to be subcritical relative to the local term under critical rescaling, leading to a loss of compactness where the fractional contribution vanishes. This paper addresses the delicate case where both operators are simultaneously critical for the same nonlinearity, making the nonlocal term a "genuine co-protagonist" rather than a perturbation.

Methodology and Framework
The analysis relies on a variational approach within the energy space $X_0(\mathbb{R}^N) = D^{1,p}_0(\mathbb{R}^N) \cap D^{s,q}_0(\mathbb{R}^N)$. Under the isocritical condition, the embedding $D^{1,p}_0(\mathbb{R}^N) \hookrightarrow D^{s,q}_0(\mathbb{R}^N)$ holds, reducing the natural energy space to $D^{1,p}_0(\mathbb{R}^N)$. However, this embedding is not compact, necessitating a refined concentration-compactness analysis.

1. Variational Structure: The authors utilize the Nehari manifold method to find ground states. They define the energy functional $J$ and the associated Nehari manifold $\mathcal{N}$. The existence proof hinges on a double-threshold concentration-compactness analysis. Unlike standard critical problems involving a single Sobolev constant, the loss of compactness here is governed by both the local best Sobolev constant $S$ and the fractional best Sobolev constant $S_f$.
2. Localization of Compactness: A key technical step involves localizing the loss of compactness on annuli. The authors prove that for a Palais-Smale sequence weakly converging to zero, the local energy, nonlocal energy, and critical mass all vanish on any annulus. This allows them to isolate the concentration of mass.
3. Rescaling: A specific rescaling argument is employed where the scaling factor preserves both the local $p$-Dirichlet energy and the nonlocal Gagliardo seminorm. This is only possible because the isocritical condition implies the exponent identity $-N + qs + q(N-p)/p = 0$.
4. Regularity and Comparison: For the decay analysis, the authors establish a weak comparison principle and a strong maximum principle for general exponents. They introduce an auxiliary space $D^{s,p,q}(\Omega)$ to handle the nonlocal operator's global nature while testing on exterior domains. A logarithmic energy estimate is derived to support these principles.
5. Harnack Inequality: A weak Harnack inequality is established for non-negative supersolutions, utilizing a De Giorgi-type iteration that balances the local term (order $r^{-p}$) and the nonlocal term (order $r^{-sq}$). The isocritical identity ensures a single Sobolev gain exponent controls both terms.

Key Contributions and Results

• Existence of Ground States (Theorem 1.1): The paper proves the existence of a non-negative radial ground state solution $u \in D^{1,p}_0(\mathbb{R}^N)$ for the problem $(P)$. The proof demonstrates that the mountain pass level coincides with the minimum on the Nehari manifold. The ground state is shown to be strictly positive via the strong maximum principle.
• Sharp Two-Sided Decay Estimates (Theorem 1.2): The authors establish the precise asymptotic decay of positive radial solutions at infinity. They prove that solutions decay exactly at the rate of the fundamental solution of the $p$-Laplacian:
  $$c_1 |x|^{-\alpha} \le U(x) \le c_2 |x|^{-\alpha}, \quad \text{where } \alpha = \frac{N-p}{p-1}.$$
  • Upper Bound: Derived using a global $L^\infty$ bound and a borderline weak-Lorentz estimate, placing the solution in the Lorentz space $L^{\eta_0, \infty}(\mathbb{R}^N)$ with $\eta_0 = \frac{(p-1)N}{N-p}$.
  • Lower Bound: Proven by constructing suitable subsolutions. The construction depends on the sign of $\alpha(q-1) - N$.
    • If $q > \frac{p(N-1)}{N-p}$, a direct regularization of the profile $|x|^{-\alpha}$ serves as a subsolution.
    • If $q < \frac{p(N-1)}{N-p}$, the authors employ a constrained minimization problem to generate a comparison function, as the raw power profile fails to be a subsolution.
• Regularity Results: The paper establishes a logarithmic energy estimate, a weak comparison principle, and a strong maximum principle for the operator $L_{p,q}$ with general exponents (not restricted to the isocritical case). Additionally, a weak Harnack inequality is proven for the isocritical regime.

Significance and Claims
The paper claims to present the first variational problem for the mixed $(p, q)$-Laplacian at "double critical growth" where the local and nonlocal best Sobolev constants jointly determine the variational level. The authors emphasize that the isocritical regime is unique because:

1. Both operators are critical for the same nonlinearity simultaneously.
2. Both operators are invariant under the same critical rescaling, preventing the "scaling collapse" of the nonlocal term observed in non-isocritical cases.
3. Despite the competition between operators, the local $p$-Laplacian dictates the large-scale behavior (decay rate) of the solutions. The nonlocal term, while essential for existence and the structure of the energy space, does not alter the asymptotic decay profile, which remains that of the fundamental solution of the $p$-Laplacian.

The authors note that while the upper bound on decay is universal (holding for general exponents), its sharpness (matching the lower bound) is specific to the isocritical regime. The paper does not claim to resolve the radial symmetry of all ground states (leaving the case $1 < p < 2$ open) but establishes the existence of radial ground states and their specific decay properties.