Ground State Normalized Solutions

• Ground state normalized solutions are defined as the least-energy standing-wave states for nonlinear Schrödinger-type PDEs with a fixed mass constraint, derived via constrained variational minimization.
• The analysis employs methods such as the Pohozaev manifold and concentration-compactness arguments to handle the challenges of mass-supercritical regimes.
• These solutions are critical in understanding the balance between dispersion and focusing nonlinear terms, with implications for threshold dynamics, blow-up, and stability.

A ground state normalized solution is a least energy standing-wave solution to nonlinear Schrödinger-type PDEs or systems, subject to a fixed mass constraint, typically in the form $\|u\|_{L^2}^2 = c^2$. In the context of mass-supercritical, fractional-sobolev-critical nonlinearities, such solutions arise from constrained variational minimization, often employing Pohozaev manifold and concentration-compactness arguments, and are characterized by critical balance between dispersion and focusing nonlinear terms. The mathematical structure and existence regime for these solutions is rich, with subtle dependence on critical exponents, problem geometry, and parameters.

1. Mathematical Setting and Definition

Consider the fractional nonlinear Schrödinger equation (NLSE): $$i\,\partial_t\psi +(-\Delta)^s\psi +\mu|\psi|^{p-2}\psi +|\psi|^{2_s^*-2}\psi =0,\quad \|\psi(t)\|_{L^2}^2 = c^2$$ where $s\in(0,1)$, $N\geq 2$, $p\in(2+\frac{4s}{N},2_s^*)$, $2_s^* = \frac{2N}{N-2s}$, and $\mu>0$. Standing-wave ansatz $\psi(t,x)=e^{i\lambda t}u(x)$ yields the stationary equation: $$(-\Delta)^s u + \lambda u = \mu |u|^{p-2}u + |u|^{2_s^*-2}u,\quad \|u\|_2=c$$ A ground state normalized solution is a minimizer of the energy functional

$$E_\mu(u) =\frac12 [u]_{H^s}^2 -\frac{\mu}{p}\|u\|_{L^p}^p -\frac{1}{2_s^*}\|u\|_{L^{2_s^*}}^{2_s^*}$$

over the $L^2$-sphere $S(c)=\{u\in H^s(\mathbb{R}^N):\|u\|_2=c\}$, subject to the Euler–Lagrange equation above with Lagrange multiplier $\lambda<0$ (Zuo et al., 2022).

2. Variational Principles and Pohozaev Manifold

The normalized problem is neither globally coercive nor bounded below—especially for mass-supercritical exponents—necessitating the use of natural constraint manifolds. The Pohozaev manifold is crucial: $P_{\mu, c} = \{u \in S(c): P_\mu(u) = 0\}$ with Pohozaev functional: $$P_\mu(u) = [u]_{H^s}^2 - \frac{N(p-2)}{2p}\mu\,\|u\|_p^p - \frac{2s}{2_s^*}\|u\|_{2_s^*}^{2_s^*}$$ Critical points of $E_\mu$ on $S(c)$ correspond to solutions of $P_\mu(u)=0$ via a scaling argument (e.g., Soave trick: $$(T_\theta u)(x) = e^{\frac{N}{2}\theta}u(e^\theta x)$$ preserves the $L^2$ norm). The constrained variational problem has mountain-pass geometry; the minimization level is

$m_\mu(c) = \inf_{u\in H^s,\,\|u\|_2=c} E_\mu(u)$

and the mountain-pass level

$$\gamma_\mu(c) = \inf_{\varphi \in C([0,1], S(c)), E_\mu(\varphi(0))>0, E_\mu(\varphi(1))<0} \max_{t\in[0,1]} E_\mu(\varphi(t))$$

is positive and less than the sharp Sobolev bubble threshold (Zuo et al., 2022).

3. Compactness and Existence Theory

A Palais–Smale sequence $\{u_n\}$ at level $\gamma_\mu(c)$ with $P_\mu(u_n)\to 0$ is bounded in $H^s$, and under strict energy inequalities (e.g., $\gamma_\mu(c) < \tfrac{s}{N}S^{N/(2s)}$, with $S$ the optimal Sobolev constant), both vanishing and dichotomy are ruled out using Brezis–Lieb splitting and the Pohozaev identity. Consequently, such a sequence converges strongly: $$u_n \rightarrow u_c \quad \text{in} \quad H^s_{\text{rad}}(\mathbb{R}^N)$$ with $u_c$ radial, positive, and satisfying the normalized ground-state Euler–Lagrange equation (Zuo et al., 2022).

4. Thresholds, Parameter Scaling, and Asymptotics

There exists a threshold $\mu_*=\mu_*(c)>0$ such that for $\mu\geq\mu_*$ the minimization level is achieved. As $\mu\to\infty$, the ground-state energy $\gamma_\mu(c)\to 0$, reminiscent of the local case $s=1$. The setting generalizes the Brézis–Nirenberg/Soave/Jeanjean critical problems, extending from $H^1$ (local Laplacian) to $H^s$ and from pure power to mixed power criticality.

5. Physical and Analytical Significance

Ground state normalized solutions correspond to the most physically relevant (least energy) states with prescribed mass (number of particles) in quantum models. Their existence indicates a balance of dispersion, focusing nonlinearity, and critical scaling. In mass-supercritical and critical regimes, such solutions play a central role in the understanding of threshold dynamics, blow-up, global existence, and instability phenomena.

Solutions on the Pohozaev manifold encode the underlying scaling invariance and are critical for handling lack of compactness; the associated functional framework and compactness arguments (profile analysis, Brezis–Lieb, Gagliardo–Nirenberg inequalities) underpin the existence theory.

6. Extensions and Related Results

• Existence and characterization of normalized ground states for the fractional nonlinear Schrödinger equation with subcritical nonlinearities, Hartree-type convolution terms, and on bounded domains have been achieved via similar variational arguments, Pohozaev constraints, and concentration–compactness methods (Feng et al., 2019, Hajaiej et al., 2023, Zhang et al., 27 Nov 2024).
• Strong instability and global existence dichotomy are firmly established in fractional and mass-supercritical cases, with sharp thresholds arising from the functional level and virial identities; the strict sign of the Pohozaev functional at ground states underpins blow-up versus global existence (Feng et al., 2019).
• Analogous frameworks apply to systems, discrete models, and critical/supercritical quasilinear equations, with ground state normalized solutions representing least energy constrained states well beyond scalar NLSE (Deng et al., 2021, Luo et al., 2023, Coster et al., 15 Nov 2024).

7. Open Directions and Context

The existence theory for ground state normalized solutions—especially in systems, nonlocal equations, and critical/supercritical regimes—remains active, with challenges in uniqueness, multiplicity, and stability. Techniques involving improved concentration–compactness, strict subadditivity, variational identities, and scaling-geometric analysis continue to generalize the classical theory to fractional, nonlocal, and multi-component contexts.

The framework set out in (Zuo et al., 2022) and related works represents the state-of-the-art for mass-supercritical, fractional Sobolev-critical NLS, elucidating both physical relevance and fundamental mathematical structure for ground state normalized solutions.