Ground State Solutions For Local-Nonlocal Shrodinger Equations In the Presence Of Two Critical Exponents

1. Problem Statement

The paper investigates the existence, positivity, and regularity of ground state solutions for a class of Schrödinger equations involving a mixed local-nonlocal operator and two critical nonlinearities. The equation under study is:

$$-\lambda \Delta u + \mu (-\Delta)^s u = f(u), \quad x \in \mathbb{R}^N$$

where:

• $N \geq 3$ and $0 < s < 1$.
• $\Delta$ is the classical Laplacian (local operator).
• $(-\Delta)^s$ is the fractional Laplacian (nonlocal operator).
• The nonlinearity $f(u)$ contains two critical exponents simultaneously: the classical Sobolev critical exponent $2^* = \frac{2N}{N-2}$ and the fractional Sobolev critical exponent $2^*_s = \frac{2N}{N-2s}$.

Specifically, the authors focus on the case where the nonlinearity is a combination of these critical terms and a subcritical term:
$$f(u) = |u|^{2^*_s - 2}u + \beta |u|^{p-2}u + |u|^{2^* - 2}u$$
with $2^*_s < p < 2^*$ and $\beta > 0$.

The Challenge:
While the existence of solutions for purely local ($\mu=0$) or purely nonlocal ($\lambda=0$) equations with critical exponents is well-understood (via Aubin-Talenti bubbles or Lieb's results), the mixed case presents a unique difficulty. Standard variational methods often fail because the presence of two distinct critical exponents creates a "double critical" situation where the compactness of the embedding is lost in a complex manner that cannot be resolved by existing techniques for single critical exponents.

2. Methodology

The authors employ a variational approach combined with novel functional analysis techniques to overcome the lack of compactness.

A. Functional Setting

The problem is set in a mixed Sobolev space $X$, defined as the completion of $C_0^\infty(\mathbb{R}^N)$ under the norm:
$$\|u\|_X^2 = \int_{\mathbb{R}^N} |\nabla u|^2 dx + \iint_{\mathbb{R}^N \times \mathbb{R}^N} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}} dx dy$$
This space embeds continuously into both $D^{1,2}(\mathbb{R}^N)$ and $D^{s,2}(\mathbb{R}^N)$, and consequently into $L^t(\mathbb{R}^N)$ for $t \in [2^*_s, 2^*]$.

B. Generalization of Lieb's Translation Theorem

The core methodological innovation is a generalized version of Lieb's translation theorem.

• Standard Issue: In critical problems, a bounded Palais-Smale (PS) sequence may lose mass at infinity (vanishing) or concentrate at a point (bubbling). Standard translation arguments usually handle one critical exponent.
• The Innovation: The authors prove that if a bounded sequence $\{u_n\}$ in $X$ has non-vanishing mass for both the $L^{2^*_s}$ and $L^{2^*}$ norms, there exists a sequence of translations $\{x_n\}$ such that the translated sequence converges strongly to a non-zero limit in $L^2_{loc}$.
• Technique: This is achieved by utilizing Morrey spaces and a delicate interpolation argument (Lemma 3.2) that links the fractional and local Sobolev norms. This allows them to control the "concentration" behavior of the sequence simultaneously for both critical exponents.

C. Variational Framework

• Mountain Pass Geometry: The energy functional $I(u)$ is shown to possess the mountain pass geometry.
• Nehari Manifold: The authors restrict the functional to the Nehari manifold $\mathcal{N} = \{u \in X \setminus \{0\} : \langle I'(u), u \rangle = 0\}$. They prove that the infimum of the energy on this manifold corresponds to the mountain pass level.
• Compactness Recovery: By establishing that the mountain pass level $c$ is strictly below a critical threshold $c^*$ (determined by the best Sobolev constants), they ensure that the PS sequence does not lose compactness due to the critical exponents.

3. Key Contributions and Results

Theorem 1.2: Existence of Ground State Solutions

• Result: There exists a threshold $\bar{\beta} > 0$ such that for any $\beta > \bar{\beta}$, the equation admits a radial ground state solution.
• Significance: This resolves the "unprecedented situation" of two critical exponents in a mixed operator setting. The result relies on the parameter $\beta$ (the coefficient of the subcritical term) being sufficiently large to perturb the energy landscape and allow the existence of a solution below the critical energy threshold.
• Positivity and Symmetry: The authors further prove that the ground state solution can be chosen to be non-negative and radially symmetric (using symmetric decreasing rearrangement inequalities).

Theorem 1.3: $L^\infty$ Regularity

• Result: Under a general growth condition on the nonlinearity $f(u)$ (specifically $|f(t)t| \leq C(t^{2^*_s} + t^{2^*})$), any non-negative solution $u$ belongs to $L^\infty(\mathbb{R}^N)$.
• Method: The proof uses a Moser iteration technique adapted for the mixed local-nonlocal operator. They construct a truncated test function $u_L$ and iterate the $L^p$ norms to establish boundedness. This is non-trivial because the nonlocal term $(-\Delta)^s$ does not satisfy the same local maximum principles as the classical Laplacian.

4. Significance and Impact

1. Resolution of a New Phenomenon: The paper addresses a gap in the literature regarding PDEs with mixed operators and multiple critical exponents. Previous methods for single critical exponents (local or nonlocal) were insufficient for this coupled scenario.
2. Novel Analytical Tool: The generalization of Lieb's translation theorem to handle simultaneous concentration of two different critical norms is a significant theoretical contribution. It provides a blueprint for tackling other problems involving mixed operators and multiple critical growths.
3. Applications: The equation models physical and biological systems where both classical diffusion (local) and long-range jumps (nonlocal) occur simultaneously, such as in population dynamics with mixed foraging strategies or pandemic spread with regional and global restrictions.
4. Regularity Theory: The establishment of $L^\infty$ bounds for solutions to mixed critical equations extends the regularity theory for fractional and mixed operators, which is crucial for further qualitative analysis (e.g., uniqueness, asymptotic behavior).

In summary, this paper successfully establishes the existence of ground states for a highly challenging class of Schrödinger equations by developing new compactness tools and refining variational methods to handle the interplay between local and nonlocal critical nonlinearities.