Existence and nonexistence of normalized solutions for nonlinear Schrödinger equation involving combined nonlinearities in bounded domain

1. Problem Statement

The paper investigates the existence and nonexistence of normalized solutions for the following nonlinear Schrödinger equation with combined power nonlinearities in a bounded smooth domain $\Omega \subset \mathbb{R}^N$:

$$\begin{cases} -\Delta u + \omega u = \mu |u|^{p-2}u + |u|^{q-2}u, & u > 0 \text{ in } \Omega, \\ u = 0, & \text{on } \partial\Omega, \\ \|u\|_{L^2(\Omega)}^2 = \rho, & \rho > 0. \end{cases}$$

Key Parameters:

• $N \ge 1$: Dimension.
• $\mu \in \mathbb{R}$: Coefficient of the lower-order nonlinearity.
• $1 < p \le q$: Exponents.
• $\omega \in \mathbb{R}$: A Lagrange multiplier arising from the mass constraint, not fixed a priori.
• $\rho$: Prescribed $L^2$-norm (mass).
• $2^* = \frac{2N}{N-2}$ (for $N \ge 3$): The Sobolev critical exponent.

The goal is to find pairs $(\omega, u)$ satisfying the equation and the mass constraint. This problem models standing waves with a fixed particle mass, a scenario distinct from the fixed-frequency problem.

2. Methodology

The authors employ a combination of variational methods, geometric analysis, and specific techniques tailored to bounded domains where standard scaling arguments fail.

• Variational Framework: Solutions are sought as critical points of the energy functional $I(u)$ on the constraint manifold $S_\rho = \{u \in H_0^1(\Omega) : \|u\|_2^2 = \rho\}$.
  $$I(u) = \frac{1}{2}\int_\Omega |\nabla u|^2 dx - \frac{\mu}{p}\int_\Omega |u|^p dx - \frac{1}{q}\int_\Omega |u|^q dx.$$
• Local Minimization: To find the first solution, the authors analyze the minimization of $I$ over subsets of $S_\rho$ defined by energy bounds ($A_\alpha$). They utilize compactness arguments and the Gagliardo-Nirenberg inequality to ensure the existence of minimizers for small mass $\rho$.
• Monotonicity Trick (Struwe's Method): To find a second solution (Mountain Pass type), the authors introduce an auxiliary parameter $\eta \in (0, 1]$ to modify the functional:
  $$I_\eta(u) = \frac{1}{2}\int_\Omega |\nabla u|^2 dx - \eta \int_\Omega \left(\frac{\mu}{p}|u|^p + \frac{1}{q}|u|^q\right) dx.$$
  By varying $\eta$, they construct a sequence of critical points. Since the domain is bounded, standard scaling invariance (used in $\mathbb{R}^N$) is lost. The monotonicity trick allows them to generate a bounded Palais-Smale (PS) sequence satisfying the Pohozaev identity.
• Pohozaev Identity & Boundedness: For star-shaped domains, the Pohozaev identity is crucial to prove the boundedness of the PS sequence in $H_0^1(\Omega)$, preventing "bubbling" (loss of compactness) at the critical level.
• Moving-Plane Method: For the nonexistence results in convex domains, the authors use the moving-plane method to establish the monotonicity of solutions near the boundary. This geometric property is used to derive contradictions for large mass $\rho$.
• Continuation Arguments: For the specific Brézis-Nirenberg case ($\mu=0, q=2^*$) in a ball, they utilize the uniqueness and radial symmetry of solutions to the classical problem to analyze the behavior of the $L^2$-norm as a function of the parameter.

3. Key Contributions and Results

A. Existence of Local Minimizers (Small Mass)

Theorem 1.1: For any $\mu \in \mathbb{R}$ and $2 < p \le q \le 2^*$, there exists a threshold $\rho_0 > 0$ such that for all $\rho \in (0, \rho_0)$, the problem admits a local minimizer solution.

• Significance: This result removes the requirement for the domain to be star-shaped (a constraint in previous works like [28]) and handles sign-changing nonlinearities ($\mu < 0$) and mass-subcritical cases.

B. Existence of a Second Solution (Mountain Pass Type)

Theorem 1.3: Assuming $\mu \ge 0$ and $\Omega$ is a star-shaped domain, if the mass $\rho$ is sufficiently small, there exist at least two positive solutions.

• Conditions:
  • $N \ge 1, 2 < p < 2 + \frac{4}{N} < q < 2^*$; OR
  • $N \ge 4, 2 < p < q = 2^*$; OR
  • $N = 3$ with specific constraints on $p$ and the inradius of $\Omega$.
• Significance: This extends previous results (e.g., [28]) to all dimensions $N \ge 3$ and establishes the existence of a second solution via the Mountain Pass geometry, overcoming the lack of compactness in bounded domains.

C. Nonexistence for Large Mass in Convex Domains

Theorem 1.5: Let $1 < p \le 2$, $q \ge \max\{2^*, 3\}$, $\mu > 0$, and $\Omega$ be a bounded convex domain. Then, for sufficiently large mass $\rho$, no positive normalized solution exists.

• Mechanism: The proof relies on the moving-plane method to show that for large $\rho$, the mass concentrates at the center, forcing the frequency $\omega$ to diverge to infinity, which contradicts the Pohozaev identity bounds.
• Significance: This provides the first nonexistence results for normalized solutions in the Sobolev critical/supercritical regime for convex domains.

D. Dichotomy for the Brézis-Nirenberg Problem in a Ball

Theorem 1.7: For the case $\mu = 0$, $q = 2^*$, and $\Omega$ being a unit ball, there exists a critical mass $\rho^*$ such that:

1. Two solutions exist if $0 < \rho < \rho^*$.
2. One solution exists if $\rho = \rho^*$.
3. No solutions exist if $\rho > \rho^*$.

• Significance: This provides a complete classification (dichotomy) for the Brézis-Nirenberg problem with prescribed mass in a ball, resolving the behavior of the $L^2$-norm of the unique solution to the classical problem as the parameter varies.

4. Significance and Impact

• Overcoming Domain Limitations: The paper successfully adapts the "monotonicity trick" (originally developed for $\mathbb{R}^N$) to bounded domains, a significant technical hurdle where scaling invariance is broken.
• Combined Nonlinearities: It addresses the complex interplay between subcritical and critical (or supercritical) nonlinearities ($\mu |u|^{p-2}u + |u|^{q-2}u$), a scenario less understood than single-power nonlinearities.
• Geometric Dependence: The results highlight the critical role of domain geometry (star-shaped vs. convex) in determining the existence and multiplicity of solutions, particularly for large mass regimes.
• Completeness: The dichotomy result for the Brézis-Nirenberg problem in a ball offers a definitive answer to the existence landscape for this specific, well-studied equation under mass constraints.

In summary, the paper advances the theory of normalized solutions by establishing robust existence criteria for small mass, proving nonexistence for large mass in convex domains, and providing a complete solution classification for the critical case in symmetric domains.