A mixed local-nonlocal Hénon problem in $\mathbb{R}^N$

Technical Summary: A Mixed Local-Nonlocal Hénon Problem in $\mathbb{R}^N$

Problem Statement
This article investigates the existence, regularity, and non-existence of positive radial solutions to a Hénon-type equation driven by a mixed local-nonlocal operator in the entire space $\mathbb{R}^N$. The problem is formulated as:
$$\begin{cases} \mathcal{L}_{s,p,\gamma} u + |x|^\beta |u|^{p-2}u = |x|^\alpha |u|^{q-2}u & \text{in } \mathbb{R}^N, \\ u > 0, \end{cases}$$
where the operator $\mathcal{L}_{s,p,\gamma}$ is defined as a convex combination of the local $p$-Laplacian and the fractional $p$-Laplacian:
$$\mathcal{L}_{s,p,\gamma} u := \gamma(-\Delta_p)u + (1-\gamma)(-\Delta_p)^s u.$$
Here, $\gamma \in [0, 1]$ is a mixing parameter, $s \in (0, 1]$ is the fractional order (with $s=1$ corresponding to the local case), and $p > 1$. The equation includes a confining potential $|x|^\beta |u|^{p-2}u$ on the left-hand side to prevent the loss of mass at infinity, a phenomenon that typically precludes existence in unbounded domains for pure Hénon equations. The right-hand side features the Hénon weight $|x|^\alpha$.

Methodology
The authors employ variational methods within a functional analytic framework tailored to radial functions.

1. Functional Setting: The problem is studied in the weighted fractional Sobolev space of radial functions, denoted $W^{s,p}_{\beta, \text{rad}}(\mathbb{R}^N)$. The energy functional associated with the problem is:
   $$J(u) = \frac{\gamma}{p}\|\nabla u\|_p^p + \frac{1-\gamma}{p}[u]_{s,p}^p + \frac{1}{p}\|u\|_{p,\beta}^p - \frac{1}{q}\|u\|_{q,\alpha}^q.$$
2. Compactness: A central technical challenge in unbounded domains is the lack of compact embeddings. The authors establish a compact embedding of the weighted radial space $W^{s,p}_{\beta, \text{rad}}(\mathbb{R}^N)$ into the weighted Lebesgue space $L^q_\alpha(\mathbb{R}^N)$. This is achieved by combining Strauss' radial lemma (which provides decay estimates for radial functions) with generalized Rother-type lemmas and specific estimates on the interaction between the weights $\alpha$ and $\beta$.
3. Existence: The existence of a non-trivial critical point for $J$ is proven using a Mountain Pass-type theorem (specifically the Brezis-Nirenberg version). The proof verifies the geometric conditions (local minimum at zero and a path where the functional becomes negative) and the Palais-Smale condition, the latter relying on the established compact embedding.
4. Regularity: Boundedness of the weak solutions is established via a De Giorgi iteration scheme adapted to the mixed operator and weighted setting.
5. Non-existence: A Pohozaev-type identity is utilized to derive a non-existence result for supercritical exponents. This involves a scaling argument and a generalized Pohozaev result for unbounded domains, requiring a weighted control of the gradient.

Key Contributions and Results

• Existence Theorem (Theorem 1.1): The authors prove the existence of a radial weak solution under the condition that the exponent $q$ lies in the subcritical range $p < q < p^*_{\gamma,s,\alpha}$, where $p^*_{\gamma,s,\alpha}$ is the critical Sobolev exponent adapted to the mixed operator and weights. Crucially, existence is guaranteed if the parameters satisfy the inequality:
  $$\alpha - \beta + (q-p)\left(1 - \frac{N}{p}\right) < 0.$$
  This result extends previous existence results for purely local ($\gamma=1$) and purely fractional ($\gamma=0$) Hénon equations to the mixed case on unbounded domains.

• Regularity (Theorem 1.2): It is shown that any radial weak solution $u$ obtained in the existence theorem is essentially bounded ($u \in L^\infty(\mathbb{R}^N)$). This is a non-trivial result given the singular nature of the weights and the nonlocal operator.

• Non-existence Threshold (Theorem 1.3): The paper establishes that $p^*_{\gamma,s,\alpha}$ serves as a sharp threshold for existence. Specifically, for $q > p^*_{\gamma,s,\alpha}$, no solutions exist in the space $\tilde{X}(\mathbb{R}^N)$ (defined as bounded solutions with a weighted gradient control $|\nabla u(x)||x| \in L^r$). This confirms that the critical exponent acts as a barrier for the existence of bounded solutions with appropriate decay.

Significance and Scope
The paper claims to extend the theory of Hénon-type equations from bounded domains and purely local or fractional settings to the more complex setting of mixed local-nonlocal operators on the whole space $\mathbb{R}^N$.

• Generalization: The results unify and generalize prior work by Ni (on bounded domains), and subsequent studies on fractional Hénon equations in $\mathbb{R}^N$ (e.g., [2, 6, 14, 20]).
• Parameter Range: The authors note that their derived condition (1.6) improves upon the parameter range obtained in [14] for the specific case of $p=2$ and $\beta=2$.
• Methodological Rigor: The work provides a rigorous functional framework for mixed operators, demonstrating that the compactness required for variational methods can be recovered in unbounded domains through the interplay of radial symmetry and weighted norms.

The paper does not propose new physical models or experimental applications but focuses strictly on the mathematical analysis of the elliptic problem, providing a theoretical foundation for the existence and properties of solutions to this class of mixed operators.