Normalized Semilinear Elliptic Problem

• Normalized semilinear elliptic problems are defined by seeking eigenfunctions and eigenvalues under an L2 constraint, leading to nonlinear eigenvalue problems via constrained minimization.
• The variational formulation utilizes energy functionals and the Pohozaev manifold to establish existence, regularity, and stability of ground state solutions in both scalar and coupled systems.
• Parametric analyticity and factorial-type mixed derivative bounds enable robust uncertainty quantification and high-dimensional integration for solutions with affine-parametric dependencies.

A normalized semilinear elliptic problem is a class of nonlinear elliptic partial differential equations (PDEs) or systems posed with $L^2$-type normalization constraints. These problems typically seek eigenfunctions and eigenvalues under a nonlinear operator structure, with normalization replacing or supplementing traditional boundary conditions. The nonlinearities can be power-type or of more general subcritical or critical nature, and the normalization requirement often arises in applications such as quantum mechanics (normalized ground states), optical physics, or mathematical biology. In the scalar case, the paradigm is to minimize an energy functional subject to an $L^2$ constraint, leading to a nonlinear eigenvalue problem. In coupled systems, as in semilinear elliptic systems, normalization conditions regulate the masses of each component. The paper of such problems involves analytical techniques for existence, regularity, parametric sensitivity, and computational properties, especially in the presence of parameter uncertainty or high-dimensional coefficient dependencies (Bahn, 2023, Li et al., 2020).

1. General Formulation

The normalized semilinear elliptic eigenvalue problem is given, on a bounded domain $\Omega \subset \mathbb{R}^d$ with $C^2$ boundary, by:

• Find $u(y;\cdot) \in H_0^1(\Omega)$, $\lambda(y)\in\mathbb{R}$ such that

$$\begin{aligned} &-\nabla\cdot(a(x)\nabla u(y,x)) + b(x,y) u(y,x) + \eta f_p(u(y,x)) = \lambda(y) u(y,x) \quad \text{in } \Omega, \ &u(y,\cdot) = 0 \quad \text{on } \partial\Omega, \ &\|u(y,\cdot)\|_{L^2(\Omega)}=1, \end{aligned}$$

where $a(x)\in C^1(\Omega)$ with $a(x) \geq a_{\min} > 0$, $b(x,y)$ is a parametric potential with affine dependence on parameters $y\in U = [-1/2,1/2]^{\mathbb{N}}$, $\eta > 0$ is fixed, and the nonlinearity is $f_p(u) = |u|^{p-1}u$ for suitable $p$ as determined by the dimension $d$ and Sobolev embedding constraints. The normalization $\|u\|_{L^2}=1$ replaces the standard $\lambda$-eigenfunction scaling ambiguity and ensures the "normalized ground state" interpretation (Bahn, 2023).

For elliptic systems, the model considered is

$$\begin{cases} -\Delta u + \lambda_1 u = \mu_1 |u|^{p-2}u + \beta r_1 |u|^{r_1-2}|v|^{r_2}u, \ -\Delta v + \lambda_2 v = \mu_2 |v|^{q-2}v + \beta r_2 |u|^{r_1}|v|^{r_2-2}v, \end{cases}$$

with normalization

$$\int_{\mathbb{R}^N} u^2 = a_1^2, \quad \int_{\mathbb{R}^N} v^2 = a_2^2,$$

where $\mu_1, \mu_2, \beta > 0$ and exponents are governed by the Sobolev critical threshold (Li et al., 2020).

2. Functional‐Analytic and Variational Structure

The variational structure is essential: solutions arise as critical points of an energy (action) functional under normalization constraints. For the scalar case, the minimization

$$E[u] = \int a|\nabla u|^2 + \int b|u|^2 + \frac{2\eta}{p+1}\int |u|^{p+1}, \quad \text{with } \|u\|_{L^2}=1,$$

leads, via Lagrange multipliers, to a normalized nonlinear eigenproblem. Strict convexity in $|u|^2$ (when applicable) ensures uniqueness up to a phase. The corresponding PDE for the coupled system includes both intra-component and inter-component nonlinear terms, and the normalization constraints define the search set $S_{a_1} \times S_{a_2}$ within $H^1(\mathbb{R}^N)\times H^1(\mathbb{R}^N)$ (Bahn, 2023, Li et al., 2020).

A key concept for systems is the Pohozaev manifold,

$$\mathcal{P}_{a_1,a_2} = \{(u,v) \in S_{a_1,a_2}: P(u,v) = 0\},$$

where $P(u,v)$ is the Pohozaev functional derived from multiplying the PDEs by $x \cdot \nabla u$, $x \cdot \nabla v$ and integrating, encoding the scaling invariance and energy geometry.

3. Existence and Regularity of Ground States

The analytic paper of normalized semilinear elliptic problems centers on the existence, uniqueness/structure, and regularity of ground states—states that minimize energy under normalization. The main results include:

• Existence is established via constrained minimization on the Pohozaev manifold or by application of the implicit function theorem in a suitable Banach space setting (Bahn, 2023, Li et al., 2020).
• Uniform bounds: There exist constants ensuring $0 < \lambda(y) \leq \bar{\Lambda}$ and $\|u(y)\|_{H^1_0} \leq \bar{U}$ uniformly over the parameter domain.
• For systems, the main theorems distinguish between $L^2$-subcritical, mixed, and supercritical regimes. Existence of positive normalized ground states in both mixed and supercritical cases relies on mass-coupling thresholds, with explicit piecewise formulas provided. In the fully supercritical case, positivity is ensured either for sufficiently large $\beta$ or when all cross-interaction exponents $r_i<2$ (Li et al., 2020).
• Every true solution lies on the Pohozaev manifold due to elliptic regularity.

4. Parametric Dependence and Analyticity

The parametric analyticity of the ground state eigenpair with respect to uncertain coefficients is a central result. Under affine-parametric dependence in $b(x,y)$ with rapidly decaying coefficients $(\|b_j\|_{L^\infty})_j \in \ell^1(\mathbb{N})$, the solution map $y \mapsto (u(y), \lambda(y))$ is complex analytic in each parameter direction and jointly so in all directions. The proof deploys:

• Reformulation via the implicit function theorem, with invertibility of the linearization ensured by the presence of a uniform linearized spectral gap between the smallest and next-smallest eigenvalues of the linearized operator.
• Hartogs’ theorem to conclude joint analyticity for countably many parameters.
• Analyticity results are pivotal for uncertainty quantification (UQ) and high-dimensional numerical quadrature algorithms (Bahn, 2023).

5. Regularity and Mixed‐Derivative Bounds

The solution map exhibits factorial-type mixed derivative bounds with respect to the parameter sequence $y$. Specifically, for any multi-index $\nu$,

$$|\partial_y^\nu \lambda(y)| \leq C_\lambda \, |\nu|! \, \beta^\nu, \qquad \|\partial_y^\nu u(y)\|_{H_0^1} \leq C_u \, |\nu|! \, \beta^\nu,$$

where $\beta_j \sim \|b_j\|_{L^\infty}$, $\beta^\nu = \prod_j \beta_j^{\nu_j}$. The derivation is by inductive differentiation of the PDE and exploits the ellipticity of the operator (parametrically uniform), careful combinatorial expansions (Faà di Bruno-type), and precise control on lower-order terms via “falling factorial” estimates. These estimates underpin the dimension-robustness of numerical approaches in high-dimensional stochastic settings (Bahn, 2023).

6. Applications: Uncertainty Quantification and High-Dimensional Integration

Given parametric uncertainty in the potential, a key application is estimating statistical moments of solution quantities,

$\mathbb{E}[Q] = \int_U Q(y)\,dy,$

where $Q(y)$ can be the principal eigenvalue or linear functionals of the ground state. Error analysis includes:

• Dimension truncation: Quantifies the impact of truncating the infinite-dimensional parameter vector $y$ to its first $s$ components, with explicit error terms $T_1(s)$, $T_2(s)$ in terms of the $\ell^q$ summability of $(\beta_j)_j$.
• Quasi-Monte Carlo (QMC) methods: For $y_j$ i.i.d. uniform, root-mean-square QMC error bounds are given in terms of the number of lattice points $N$ and a critical exponent $\alpha$, with convergence rates independent of nominal parameter dimension. The total error, combining truncation and QMC integration, remains dimension-independent (Bahn, 2023).

7. Thresholds, Compactness, and Geometry in Elliptic Systems

For normalized elliptic systems, the role of threshold functions $T(a_1,a_2)$ is fundamental for existence results. Explicit formulas for $T(a_1,a_2)$ involve the exponents and coupling constants, distinguishing cases according to whether the nonlinearity is subcritical, critical, or supercritical with respect to $L^2$-scaling. The geometry of the energy landscape, as explored via fiber maps preserving normalization, determines when constrained minimization yields ground states. Radial symmetry, compactness via the Strauss embedding, and careful energy-level estimates are deployed to rule out pathological minimizing sequences (vanishing, dichotomy). The results refine and extend work on both single and coupled nonlinear Schrödinger equations (Li et al., 2020).