The new observations about the parameter-dependent Schrödinger-Poisson system

1. Problem Statement

The paper investigates the existence and asymptotic behavior of non-trivial weak solutions for the parametric Schrödinger-Poisson system in $\mathbb{R}^3$:
$$\begin{cases} -\Delta u + V(x)u - \lambda \phi u = f(u) & \text{in } \mathbb{R}^3, \\ -\Delta \phi = u^2 & \text{in } \mathbb{R}^3, \end{cases}$$
where:

• $u$ represents the wave function and $\phi$ the electrostatic potential.
• $\lambda > 0$ is a parameter (related to the frequency of standing waves).
• $V(x)$ is a given potential function.
• $f(u)$ is a nonlinear term satisfying specific growth conditions.

The system arises from the study of standing waves in the time-dependent Schrödinger equation coupled with a Poisson equation. The authors focus on two distinct scenarios regarding the potential $V(x)$ and the parameter $\lambda$:

1. Case 1: $V(x)$ is positive, radial, and bounded away from zero ($V(x) \ge V_0 > 0$).
2. Case 2: $V(x)$ is coercive but sign-changing (allowing the Schrödinger operator $-\Delta + V$ to be indefinite), and $\lambda$ is sufficiently large.

2. Methodology

The authors employ variational methods combined with novel analytical techniques to overcome difficulties associated with the lack of compactness and the indefinite nature of the operator in Case 2.

A. Reformulation and Truncation

To handle the behavior of the nonlinearity $f(u)$, the authors introduce a scaling transformation $\lambda = \epsilon^{-2}$ and a truncation function. They define a modified functional $I_\epsilon$ (and $I_{\epsilon, R}$ for the sign-changing case) where the nonlinearity is truncated for large arguments. This ensures the functional is well-defined on the working Hilbert spaces ($H^1_r(\mathbb{R}^3)$ for radial potentials and $H^1_V(\mathbb{R}^3)$ for sign-changing potentials).

B. Key Technical Innovation: Poisson Equation Properties

A central novelty of the paper is the utilization of a specific inequality derived from the properties of the Poisson equation solution $\phi_u$:
$$\int_{\mathbb{R}^3} \phi_u u^2 dx = \int_{\mathbb{R}^3} |\nabla \phi_u|^2 dx \ge \int_{\mathbb{R}^3} |u|^3 dx - \frac{1}{4} \int_{\mathbb{R}^3} |\nabla u|^2 dx.$$
This inequality is crucial for proving the boundedness of Palais-Smale (PS) sequences (or Cerami sequences) without relying on the non-degeneracy of solutions to the limiting equation, a technique previously required in similar works (e.g., Jeong and Seok [12]).

C. Case-Specific Approaches

• Case 1 (Positive Potential): The energy functional satisfies the geometric conditions of the Mountain Pass Theorem. The authors prove the existence of a critical point by establishing the Mountain Pass geometry and verifying the (PS) condition using the new Poisson inequality and Moser iteration to remove the truncation.
• Case 2 (Sign-Changing Potential): Since $V(x)$ changes sign, the operator is indefinite, and $u=0$ is not a local minimizer. The authors use:
  • Local Linking: To analyze the behavior of the functional near zero with respect to the decomposition of the space into negative and positive eigenspaces.
  • Morse Theory: Specifically, computing critical groups at zero ($C_q(I, 0)$) and at infinity ($C_q(I, \infty)$). They show that $C_d(I, 0) \neq 0$ (where $d$ is the dimension of the negative space) while $C_q(I, \infty) = 0$ for all $q$. This topological discrepancy guarantees the existence of a non-trivial critical point.

3. Key Contributions and Innovations

1. Relaxed Conditions on Nonlinearity: Unlike previous studies (e.g., Mugnai [20], Jeong and Seok [12]) that required growth conditions at infinity (such as the Ambrosetti-Rabinowitz condition or specific bounds on $f(t)$ as $|t| \to \infty$), this paper only assumes that $f$ satisfies a super-linear growth condition at the origin ($\lim_{t \to 0} f(t)/t = 0$). The behavior of $f$ at infinity is not restricted.
2. New Boundedness Argument: The proof of the boundedness of (PS) sequences relies on a new property of the Poisson equation solution rather than the non-degeneracy of the limiting problem. This allows the removal of the Ambrosetti-Rabinowitz (AR) condition.
3. Sign-Changing Potentials with $\lambda > 0$: While many works study sign-changing potentials with $\lambda < 0$, this paper establishes existence results for $\lambda > 0$ (large) with sign-changing $V(x)$, a significantly more difficult scenario due to the indefiniteness of the operator.
4. Asymptotic Analysis: The paper provides a rigorous analysis of the asymptotic behavior of solutions as $\lambda \to \infty$.

4. Main Results

Theorem 1.1 (Positive Radial Potential)

If $V(x)$ is radial, positive, and bounded below, and $f$ satisfies the super-linear condition at the origin, there exists a threshold $\lambda^* > 0$ such that for all $\lambda \ge \lambda^*$, the system admits a non-trivial weak solution.

Theorem 1.3 (Sign-Changing Potential)

If $V(x)$ is bounded from below, sign-changing, and satisfies specific measure-theoretic conditions (ensuring the spectrum is discrete), and $f$ satisfies local conditions near zero, there exists a threshold $\lambda^{**} > 0$ such that for all $\lambda \ge \lambda^{**}$, the system admits a non-trivial weak solution. This result is obtained via local linking and Morse theory.

Theorem 1.6 (Asymptotic Behavior)

As $\lambda \to \infty$ (equivalently $\epsilon \to 0$):

1. The solution $u_\lambda$ converges strongly to $0$ in $H^1_r(\mathbb{R}^3)$.
2. The scaled solution $\sqrt{\lambda}u_\lambda$ converges strongly to a non-trivial weak solution $u_0$ of the Choquard equation:
   $$-\Delta u + V(x)u = \left( \frac{1}{4\pi} \int_{\mathbb{R}^3} \frac{u^2(y)}{|x-y|} dy \right) u.$$

5. Significance

This paper advances the theory of Schrödinger-Poisson systems by:

• Generalizing Existence Conditions: It demonstrates that for large parameters $\lambda$, the existence of solutions depends primarily on the local behavior of the nonlinearity near the origin, rather than its global growth.
• Methodological Advancement: It introduces a robust technique for handling indefinite operators and unbounded nonlinearities without relying on the restrictive Ambrosetti-Rabinowitz condition or non-degeneracy assumptions.
• Bridging Gaps: It connects the parametric Schrödinger-Poisson system to the Choquard equation in the limit of large parameters, providing insight into the physical behavior of the system under strong coupling.

The methodology developed is noted by the authors as adaptable to other related problems involving the existence of solutions for Schrödinger-Poisson systems with indefinite potentials.