Variational problems related to self-similar solutions of Hardy-Sobolev heat equation in RN

Technical Summary: Variational Problems Related to Self-Similar Solutions of Hardy-Sobolev Heat Equation in $\mathbb{R}^N$

Problem Statement
This paper investigates the existence, multiplicity, and nonexistence of self-similar solutions for the semilinear parabolic equation with a singular Hardy-Sobolev potential in $\mathbb{R}^N$:
$$u_t - \Delta u = \frac{|u|^{q-2}u}{|x|^s}, \quad \text{in } \mathbb{R}^N \times (0, \infty),$$
where $2 < q \leq 2^*(s) \equiv \frac{2N-2s}{N-2}$ and $0 \leq s < 2$. The study focuses on converting this parabolic problem into an equivalent elliptic problem via a self-similar transformation. The transformed equation is:
$$-\Delta v - \frac{1}{2}y \cdot \nabla v = \alpha v + \frac{|v|^{q-2}v}{|y|^s}, \quad \text{in } \mathbb{R}^N,$$
where $\alpha = \frac{2-s}{2q-4}$. The primary objective is to analyze the variational structure of this elliptic equation, specifically addressing the subcritical ($q < 2^*(s)$) and critical ($q = 2^*(s)$) cases under the influence of the singular term $|y|^{-s}$ and the Gaussian weight $K(y) = e^{|y|^2/4}$.

Methodology
The authors employ a combination of functional analysis, weighted inequalities, and variational methods within a specific Hilbert space framework.

1. Functional Setting: The problem is formulated in a Hilbert space $X$, defined as the completion of $C_c^\infty(\mathbb{R}^N)$ with respect to the norm induced by the inner product $(v_1, v_2)_K = \int_{\mathbb{R}^N} (\nabla v_1 \cdot \nabla v_2) K(y) dy$. The weak solutions correspond to critical points of the energy functional $E_K(v)$.
2. Weighted Inequalities: A foundational step involves establishing the weighted Hardy inequality and the weighted Sobolev-Hardy inequality. These inequalities are crucial for proving the continuity and compactness of the relevant embeddings $X \hookrightarrow L^q_K(\mathbb{R}^N, |y|^{-s})$.
3. Variational Techniques:
   • Subcritical Case: The Fountain Theorem is utilized to establish the existence of infinitely many solutions. This requires verifying the geometric conditions of the theorem and the Palais-Smale (PS) condition.
   • Critical Case: The existence of solutions relies on the concentration-compactness principle and the analysis of the best constant $S_{K,\alpha}$ associated with the weighted Sobolev-Hardy inequality. The authors compare $S_{K,\alpha}$ with the standard best constant $S^0$ (and $S_{K,0}$) to ensure the infimum is achieved.
   • Nonexistence: The Pohozaev identity is applied to derive conditions under which no nontrivial solutions exist.
4. Alternative Proof: For the critical case, an alternative proof is provided using the Mountain Pass Theorem, relying on a specific energy level threshold to verify the (PS) condition.

Key Contributions and Results

• Inequalities: The paper rigorously proves the K-Hardy inequality and the K-Sobolev-Hardy inequality, establishing the necessary compactness properties for the embedding of the weighted space $X$ into weighted Lebesgue spaces.
• Subcritical Case ($2 < q < 2^*(s)$):
  • Theorem 3.1: For any $\alpha \in \mathbb{R}$, the problem admits infinitely many solutions $\{\pm v_k\}_{k=1}^\infty$ such that the energy $E_K(\pm v_k) \to +\infty$ as $k \to \infty$. This implies the existence of infinitely many self-similar solutions for the original heat equation.
• Critical Case ($q = 2^*(s)$):
  • Nonexistence: Theorem 4.1 establishes that if $\alpha \leq N/4$ (for $N \geq 3$), the only solution to the problem is the trivial solution $v \equiv 0$. This is proven via the Pohozaev identity.
  • Existence:
    • For $N \geq 4$ and $\alpha \in (N/4, N/2)$, there exists at least one nontrivial solution (Theorem 4.2).
    • For $N = 3$ and $\alpha \in (1, 3/2)$, there exists at least one nontrivial solution.
    • The paper also characterizes ranges where positive solutions do not exist (e.g., $\alpha \leq N/4$ or $\alpha \geq N/2$ for $N \geq 4$).
  • Optimality of Constants: The authors demonstrate that for specific ranges of $\alpha$, the infimum of the quotient functional $Q_{K,\alpha}$ is strictly less than the standard best constant $S^0$, ensuring the attainment of the minimum and thus the existence of a solution.
• Alternative Approach: Section 5 provides a second proof for the existence results in the critical case using the Mountain Pass geometry, confirming the robustness of the existence claims under the derived energy constraints.

Significance and Claims
The paper claims to extend existing literature on self-similar solutions of heat equations (specifically the work of Escobedo, Kavian, and Tan regarding the case $s=0$) to the more complex scenario involving singular Hardy-Sobolev terms ($s > 0$).

The significance lies in:

1. Completing the Dichotomy: By establishing both existence and nonexistence results based on the parameter $\alpha$ and dimension $N$, the work clarifies the solvability landscape for the transformed elliptic equation.
2. Handling Singularities: The derivation of weighted inequalities and the application of variational methods in the presence of the singular term $|y|^{-s}$ and the Gaussian weight $K(y)$ provide a rigorous framework for analyzing such singular evolution problems.
3. Multiplicity: The proof of infinitely many solutions in the subcritical case highlights the rich structure of the solution set for these types of parabolic equations.

The authors maintain a modest tone, focusing strictly on the mathematical derivation of existence, multiplicity, and nonexistence theorems without extrapolating to specific physical applications beyond the general characterization of diffusion and singular evolution phenomena mentioned in the introduction. The results are presented as rigorous extensions of classical theories to the Hardy-Sobolev setting.