Fučik spectrum for the operator with rapidly increasing weight and applications

1. Problem Statement

The paper investigates the Fučik spectrum for a specific differential operator defined on the entire Euclidean space $\mathbb{R}^N$ ($N \geq 1$). The operator $L$ is given by:
$$Lu := -\Delta u - \frac{1}{2}(x \cdot \nabla u)$$
The study focuses on the set $\Sigma \subset \mathbb{R}^2$ of pairs $(\alpha, \beta)$ such that the following nonlinear eigenvalue problem admits a non-trivial solution $u \in X$:
$$\begin{cases} Lu = \alpha u^+ - \beta u^-, & \text{in } \mathbb{R}^N, \\ u \in X, \end{cases}$$
where $u^+ = \max(u, 0)$, $u^- = \max(-u, 0)$, and $X$ is a weighted Sobolev space (defined below).

Key Challenges:

• Unbounded Domain: Unlike classical Fučik spectrum problems defined on bounded domains $\Omega$, this problem is set on $\mathbb{R}^N$. This prevents the direct application of standard compactness arguments and estimates used in bounded settings.
• Rapidly Increasing Weight: The operator $L$ is associated with the weight function $K(x) = e^{|x|^2/4}$. This weight arises naturally when seeking self-similar solutions to the heat equation $v_t - \Delta v = |v|^{p-2}v$. The rapid growth of $K(x)$ at infinity requires specialized functional analytic tools.

2. Methodology

The authors employ variational methods and minimax theory to construct the spectrum and analyze its properties.

• Functional Setting:

  • The space $X$ is the completion of $C_c^\infty(\mathbb{R}^N)$ under the norm $\|u\| = \left( \int_{\mathbb{R}^N} |\nabla u|^2 K(x) dx \right)^{1/2}$.
  • The weight $K(x) = e^{|x|^2/4}$ allows the operator to be written in divergence form: $Lu = -\frac{1}{K}\nabla \cdot (K \nabla u)$.
  • Embedding Theorems: The authors utilize results from Escobedo and Kavian [14] to establish that $X$ embeds continuously into weighted Lebesgue spaces $L^q_K$ for $q \in [2, 2^*]$ and compactly for $q \in [2, 2^*)$. This compactness is crucial for handling the unbounded domain.

• Variational Formulation:

  • Weak solutions correspond to critical points of the functional:
    $$J(u) = \frac{1}{2}\|u\|^2 - \frac{\alpha}{2}\|u^+\|_{K,2}^2 - \frac{\beta}{2}\|u^-\|_{K,2}^2$$
  • To construct the spectrum, the authors define a constrained functional $\tilde{I}_p$ on the manifold $P = \{u \in X : \|u\|_{K,2} = 1\}$:
    $$\tilde{I}_p(u) = \|u\|^2 - p\|u^+\|_{K,2}^2$$
  • They utilize the minimax principle (specifically the Mountain Pass geometry) to find critical values $c(p)$ of $\tilde{I}_p$.

• Topological Tools:

  • The proof relies on the Palais-Smale (PS) condition (or Cerami condition) to ensure the convergence of critical sequences.
  • Ekeland's Variational Principle is used to construct paths and prove the existence of critical points.
  • Topological properties of the manifold $P$ (local arcwise connectedness) are used to construct paths connecting the first eigenfunction $\phi_1$ and its negative $-\phi_1$.

3. Key Contributions and Results

A. Construction of the First Nontrivial Curve

The authors successfully construct three curves in the Fučik spectrum $\Sigma$:

1. The vertical line: $\{\lambda_1\} \times \mathbb{R}$
2. The horizontal line: $\mathbb{R} \times \{\lambda_1\}$
3. The first nontrivial curve $C$: Defined as $C = \{(p + c(p), c(p)) : p \in \mathbb{R}\}$, where $c(p)$ is a critical value obtained via the minimax formula.

B. Properties of the Curve $C$

The paper establishes several rigorous properties for the curve $C$:

• Lipschitz Continuity: The map $p \mapsto (p+c(p), c(p))$ is Lipschitz continuous on $\mathbb{R}$.
• Strict Monotonicity: The curve is strictly decreasing. Specifically, if $p_1 < p_2$, then $c(p_1) > c(p_2)$ and $p_1 + c(p_1) < p_2 + c(p_2)$.
• Asymptotic Behavior: As $p \to \infty$, $c(p) \to \lambda_1$ (the first eigenvalue of $L$).
• Symmetry: The spectrum exhibits symmetry properties derived from the relation $\tilde{I}_{-p}(-u) = \tilde{I}_p(u) + p$.

C. Characterization of Sign-Changing Solutions

• The authors prove that any sign-changing solution corresponds to a point $(\alpha, \beta)$ strictly in the region $\alpha > \lambda_1$ and $\beta > \lambda_1$ (above the "broken line" formed by the first eigenvalue axes).
• They demonstrate that the first eigenfunction $\phi_1$ corresponds to the point $(\lambda_1, \lambda_1)$, and any solution on the lines $\{\lambda_1\} \times \mathbb{R}$ or $\mathbb{R} \times \{\lambda_1\}$ must be a signed (non-sign-changing) solution.

D. Application: Multiplicity of Solutions

The theoretical results on the Fučik spectrum are applied to the nonlinear problem:
$$\begin{cases} -\Delta u - \frac{1}{2}(x \cdot \nabla u) = f(x, u), & \text{in } \mathbb{R}^N, \\ u \in X, \end{cases}$$
where $f(x, u)$ is asymptotically linear at both zero and infinity.

• Conditions: The nonlinearity $f$ satisfies growth conditions and limits $f_0 = \lim_{t\to 0} f/t$ and $f_\infty = \lim_{t\to \infty} f/t$ such that $\lambda_1 < f_0, f_\infty < \infty$.
• Result: Under these conditions, the problem admits at least two positive solutions.
  • One solution is obtained via a local minimization argument (near zero).
  • The second solution is obtained via the Mountain Pass Theorem, utilizing the geometry provided by the Fučik spectrum analysis (specifically, the crossing of the spectrum by the asymptotic slopes of $f$).

4. Significance

• Extension to Unbounded Domains: This work extends the theory of Fučik spectra from bounded domains to the whole space $\mathbb{R}^N$ with rapidly increasing weights, a setting previously less explored due to technical difficulties with compactness.
• Self-Similar Solutions: The operator $L$ is intrinsically linked to the self-similar solutions of the heat equation. Understanding its spectrum provides deeper insight into the long-time behavior and stability of such solutions.
• New Multiplicity Results: The paper provides a robust framework for proving the existence of multiple solutions for asymptotically linear problems in weighted spaces, overcoming the lack of compactness inherent in unbounded domains by leveraging the specific decay properties of the weight $K(x)$.
• Methodological Rigor: The detailed construction of the first nontrivial curve and the proof of its Lipschitz continuity and asymptotic behavior in this specific weighted setting serve as a template for similar problems involving non-standard operators.

In summary, Bai and Fang provide a comprehensive analysis of the Fučik spectrum for a weighted operator on $\mathbb{R}^N$, establishing the existence and properties of the first nontrivial curve and applying these findings to guarantee the multiplicity of positive solutions for related nonlinear elliptic equations.