Nonlinear Schrödinger equations with a critical, inverse-square potential

Technical Summary: Nonlinear Schrödinger Equations with a Critical, Inverse-Square Potential

Problem Statement
The paper investigates the existence of standing wave solutions for the nonlinear Schrödinger equation (NLS) featuring a critical inverse-square potential (Hardy-type potential) and a periodic nonlinearity. The specific equation under study is:
$$-\Delta u + V(x)u - \frac{(N-2)^2}{4|x|^2}u = f(x, u), \quad x \in \mathbb{R}^N \setminus \{0\}$$
where $N \ge 3$. The potential $V(x)$ and the nonlinearity $f(x, u)$ are assumed to be $\mathbb{Z}^N$-periodic. The coefficient of the singular term is critical, equal to the optimal constant in the Hardy inequality, $\mu = \frac{(N-2)^2}{4}$. The nonlinearity $f$ satisfies Carathéodory conditions, is superlinear, and exhibits subcritical growth (power $p \in (2, 2^*)$, where $2^* = \frac{2N}{N-2}$).

Methodology and Functional Setting
The authors employ variational methods to find critical points of the associated energy functional. A primary challenge addressed is the lack of translation invariance caused by the singular potential term $\frac{1}{|x|^2}$, which prevents the direct application of standard compactness arguments used in periodic NLS problems.

1. Nonstandard Functional Space ($X^1(\mathbb{R}^N)$):
   Due to the critical coefficient in the Hardy inequality, the standard $H^1(\mathbb{R}^N)$ norm is not equivalent to the natural energy norm of the problem. The authors define the space $X^1(\mathbb{R}^N)$ as the completion of $H^1(\mathbb{R}^N)$ under the norm:
   $$\|u\|_{X^1}^2 = \int_{\mathbb{R}^N} \left( |\nabla u|^2 - \frac{(N-2)^2}{4|x|^2}u^2 + u^2 \right) dx$$
   This space is a Hilbert space that embeds continuously into $H^s(\mathbb{R}^N)$ for $s \in (0, 1)$ and into $L^t(\mathbb{R}^N)$ for $t \in [2, 2^*)$. The authors utilize a specific norm $\|u\|^2 = [u]^2 + \int V(x)u^2 dx$, which is equivalent to $\|\cdot\|_{X^1}$ under the assumption that $V$ has a positive essential infimum.

2. Variational Framework:
   The energy functional $J: X^1(\mathbb{R}^N) \to \mathbb{R}$ is defined as:
   $$J(u) = \frac{1}{2}\|u\|^2 - \int_{\mathbb{R}^N} F(x, u) dx$$
   where $F$ is the primitive of $f$. The authors utilize an abstract theorem (based on [1]) to generate Cerami sequences on the Nehari manifold. The assumptions on $f$ (superlinearity, subcritical growth, and monotonicity of $f(x,u)/|u|$) ensure the functional satisfies the geometric conditions required for the existence of such sequences.

3. Profile Decomposition:
   Since the Palais-Smale and Cerami compactness conditions are not satisfied due to the lack of translation invariance in $X^1(\mathbb{R}^N)$, the authors develop a new profile decomposition adapted to this specific structure.

   • They establish that any bounded Cerami sequence $(u_n)$ in $X^1(\mathbb{R}^N)$ can be decomposed into a weak limit $u_0$ (a solution to the original equation) and a finite number of "bubbles" $u_i$ (solutions to the limiting equation without the singular potential, defined on $H^1(\mathbb{R}^N)$).
   • Crucially, they prove that the singular potential term vanishes in the limit for the translated bubbles. By comparing the energy levels of the original functional $J$ and the limiting functional $J_0$ (associated with the equation $-\Delta u + V(x)u = f(x,u)$), they demonstrate that nontrivial bubbles cannot exist at the ground state energy level.

Key Contributions and Results

• Existence of Ground State: The main result (Theorem 1.1) establishes the existence of a ground state solution to equation (1.2) under assumptions (V) and (F1)–(F4).
• New Profile Decomposition: The paper introduces a technically complex profile decomposition theorem (Theorem 4.4) specifically tailored for the space $X^1(\mathbb{R}^N)$. This decomposition handles the non-translation-invariant nature of the singular potential by showing that any weak limit of a translated sequence must be a critical point of the limiting functional defined only on $H^1(\mathbb{R}^N)$.
• Compactness Recovery: By comparing energy levels ($c$ for the original problem and $c_0$ for the limiting problem), the authors show that $c < c_0$. This inequality forces the number of nontrivial bubbles ($m$) in the profile decomposition to be zero, thereby recovering strong convergence of the Cerami sequence to a ground state.

Significance and Claims
The paper claims to resolve the existence problem for NLS equations with a critical inverse-square potential in a periodic setting, a case that introduces significant difficulties compared to subcritical cases (where $\mu < \frac{(N-2)^2}{4}$). The authors note that the critical constant destroys the equivalence of the natural norm with the classical $H^1$ norm and breaks translation invariance, rendering standard compactness tools inapplicable.

The significance lies in the development of the new profile decomposition adapted to the non-translation-invariant structure of $X^1(\mathbb{R}^N)$. This approach allows the authors to exclude nontrivial weak limits by leveraging the energy gap between the singular and non-singular problems. The work builds upon and extends previous studies on subcritical Hardy potentials (e.g., [13], [15]) and the study of the Berestycki–Lions equation in this specific functional setting (e.g., [3]). The paper does not propose new physical applications but rather provides a rigorous mathematical framework for solving this specific class of singular elliptic problems.