Periodic solutions for p(t)-Lienard equations with a singular nonlinearity of attractive type

1. Problem Statement

The paper investigates the existence of $T$-periodic solutions for a class of second-order differential equations involving the $p(t)$-Laplacian operator and a singular nonlinearity of attractive type. The specific boundary value problem (BVP) is given by:

$$\begin{cases} (\phi_{p(t)}(u'(t)))' + f(u(t))u'(t) + g(u(t)) = h(t), & t \in [0, T] \\ u(0) - u(T) = 0, \quad u'(0) - u'(T) = 0 \end{cases}$$

Key Components:

• Operator: $\phi_{p(t)}(x) = |x|^{p(t)-2}x$, where $p: [0, T] \to (1, \infty)$ is a continuous function with $p(0)=p(T)$. This generalizes the standard $p$-Laplacian to a variable exponent case.
• Nonlinearities:
  • $f: [0, \infty) \to \mathbb{R}$ is continuous (damping term).
  • $g: (0, \infty) \to \mathbb{R}$ is continuous and exhibits an attractive singularity at zero, meaning $\lim_{x \to 0^+} g(x) = +\infty$.
  • $h \in C([0, T]; \mathbb{R})$ is a periodic forcing term.
• Domain: Solutions are sought in the space $C^1_T$ (continuously differentiable functions satisfying periodic boundary conditions) such that $u(t) > 0$ for all $t$.

2. Methodology

The authors employ a topological approach combining Mawhin's continuation theorem with the method of lower and upper solutions.

A. Continuation Theorem (Mawhin's Degree Theory)

The core existence tool is a continuation theorem (Theorem 2.3) adapted from recent work [4] for mappings involving the $p(t)$-Laplacian. To apply this, the authors must verify three conditions for a bounded open set $\Omega$:

1. A priori bounds: No solution exists on the boundary $\partial \Omega$ for the homotopy problem $(\phi_{p(t)}(u'))' = \lambda \ell(t, u, u')$ for $\lambda \in (0, 1)$.
2. Non-resonance: The averaged equation $L(a) = \frac{1}{T}\int_0^T \ell(t, a, 0) dt = 0$ has no solution on $\partial \Omega \cap \mathbb{R}$.
3. Degree condition: The Brouwer degree $\deg_B[L, \Omega \cap \mathbb{R}, 0] \neq 0$.

B. Auxiliary Problems and A Priori Estimates

To establish the necessary bounds, the authors analyze auxiliary problems:

• Perturbed Liénard Equation: They first study $(\phi_{p(t)}(u'))' + \theta(u)u' - \varepsilon u = e(t)$ with $\varepsilon > 0$ and mean-zero $e(t)$. Using energy estimates and Sobolev inequalities, they derive uniform bounds on $\|u\|_\infty$ and $\|u'\|_\infty$ independent of $\varepsilon$ (Theorem 2.4).
• Limit Case ($\varepsilon = 0$): By taking the limit as $\varepsilon \to 0$, they extend existence results to the case without the linear damping term (Theorem 2.7).
• Uniform Bounds: Proposition 2.8 establishes that solutions to $(\phi_{p(t)}(u'))' + \theta(c+u)u' = e(t)$ satisfy a uniform bound $\|u\|_\infty \leq K$ independent of the shift $c$ and the function $\theta$. This is crucial for constructing upper solutions.

C. Method of Lower and Upper Solutions

The authors construct a direct proof (Theorem 3.3) for the existence of solutions between a lower solution $\alpha$ and an upper solution $\beta$ ($\alpha \leq \beta$).

• Lower Solution ($\alpha$): A function satisfying the differential inequality $\geq h(t)$ with appropriate boundary derivative conditions.
• Upper Solution ($\beta$): A function satisfying the differential inequality $\leq h(t)$.
• Truncation: They define a modified problem using a truncation function $\gamma(t, x)$ that forces the solution to stay within $[\alpha, \beta]$. The existence of a solution to this modified problem is guaranteed by the continuation theorem, and a contradiction argument proves that this solution naturally satisfies $\alpha \leq u \leq \beta$.

3. Key Contributions

1. Extension to Variable Exponents: The paper generalizes previous results (which were mostly for constant $p$ or $p=2$) to the $p(t)$-Laplacian case. This introduces significant technical challenges due to the lack of homogeneity and the variable growth rate of the operator.
2. Handling Attractive Singularities: The authors successfully address the case where the restoring force $g(u)$ becomes infinite as $u \to 0^+$. This is physically relevant for models involving gravitational or electrostatic forces where collision (zero distance) must be avoided.
3. Direct Proof of Lower/Upper Solutions: Unlike previous works that relied on variational methods (which are not applicable here due to the lack of a variational structure for the $p(t)$-Liénard equation with damping), the authors provide a direct topological proof of the lower/upper solution method tailored for the $p(t)$-Laplacian.
4. Necessary and Sufficient Conditions: They derive precise conditions on the forcing term $h(t)$ for the existence of positive periodic solutions in the singular case.

4. Main Results

Theorem 4.1 (Main Existence Result):
Problem (1.1) has at least one positive $T$-periodic solution if:

1. There exists a constant $\alpha > 0$ such that $g(\alpha) \geq \max_{t \in [0, T]} h(t)$.
2. $\limsup_{x \to +\infty} g(x) < \bar{h}$ (where $\bar{h}$ is the mean value of $h$).

Corollary 4.3 (Singular Case Characterization):
If $g(x) \to +\infty$ as $x \to 0^+$ and $g(x) \to 0$ as $x \to +\infty$ (attractive singularity), then the problem has a positive solution if and only if the mean value of the forcing term is positive ($\bar{h} > 0$).

Example 4.4:
The result applies to specific singular forms like $g(u) = \sigma u^{-\mu}$ (with $\sigma, \mu > 0$), confirming that a solution exists iff $\bar{h} > 0$.

5. Significance and Limitations

• Significance: This work bridges the gap between classical constant-exponent Liénard equations and the more complex variable-exponent ($p(t)$) setting. It demonstrates that topological methods (degree theory) remain robust even when the differential operator lacks standard homogeneity and when singularities are present. It provides a rigorous framework for analyzing physical systems with variable material properties (modeled by $p(t)$) and singular forces.
• Limitations:
  • The results are currently restricted to attractive singularities ($g(x) \to +\infty$). The authors explicitly state that the extension to repulsive singularities ($g(x) \to -\infty$) remains an open problem.
  • The approach relies on the continuity of $p(t)$ and the condition $p(0)=p(T)$.
  • Variational methods are noted as inapplicable here, necessitating the use of more sophisticated topological tools and variable exponent Sobolev spaces for the analysis.

In summary, the paper establishes a robust existence theory for periodic solutions in a challenging class of nonlinear differential equations, successfully combining degree theory with the method of lower and upper solutions to handle both variable exponents and singular nonlinearities.