Positive Solutions for a One-Dimensional Minkowski-Curvature Equation with a Sign-Changing Nonlinearity under Mixed Boundary Conditions

Technical Summary: Positive Solutions for a One-Dimensional Minkowski-Curvature Equation with Sign-Changing Nonlinearity

Problem Statement
The paper investigates the existence of positive solutions for a one-dimensional Minkowski-curvature equation subject to mixed Neumann–Dirichlet boundary conditions. The specific boundary value problem (BVP) is defined on an interval $I = [a, b]$ as:
$$\begin{cases} -\left( \frac{u'(t)}{\sqrt{1 - (u'(t))^2}} \right)' = f(t, u(t)), & t \in (a, b), \\ u'(a) = 0, \\ u(b) = 0. \end{cases}$$
The primary constraints are that the solution must satisfy the spacelike condition $\|u'\|_\infty < 1$ and be positive ($u(t) > 0$ for $a \le t < b$). A critical feature of the problem is that the nonlinearity $f: I \times [0, +\infty) \to \mathbb{R}$ is continuous but sign-changing (it may take negative values) and is not required to be separable. This distinguishes the work from previous studies that often assumed nonnegative nonlinearities or treated sign-changing terms in structurally different settings (e.g., periodic or autonomous cases).

Methodology
The author employs a topological approach based on fixed point index theory within a cone of a Banach space. The methodology addresses two specific challenges inherent to the Minkowski-curvature operator and sign-changing nonlinearities:

1. Global Definition and Slope Control: The operator involves the term $(1 - (u')^2)^{-3/2}$, which is undefined if $|u'| \ge 1$. To circumvent this, the author introduces a globally defined auxiliary equation using a continuous extension $h(p)$ of the factor $(1-p^2)^{3/2}$. They utilize a "first-contact argument" to prove that any nonnegative solution to this auxiliary equation automatically satisfies $\|u'\|_\infty < 1$, thereby recovering a solution to the original problem.
2. Handling Sign-Changing Nonlinearities: Standard cone expansion-compression theorems often fail when $f$ changes sign, as the associated integral operator may not preserve a positive cone. The author overcomes this by:
   • Imposing a one-sided linear bound: $f(t, s) \ge -\beta(t)s$ for some nonnegative continuous function $\beta$.
   • Constructing a nonnegative linear shift to transform the problem into one involving the operator $-u'' + \beta(t)u$.
   • Utilizing the Green's function $G_\beta(t, s)$ associated with the shifted linear problem to define an invariant cone $K$ in the space $C^1(I)$.

Key Contributions and Results
The paper establishes the existence of at least one positive solution under the following conditions:

• Hypothesis (H0): A lower bound condition $f(t, s) \ge -\beta(t)s$ controlling the negative part of the nonlinearity.
• Hypothesis (H1): A lower comparison near zero involving a weight function $b_0(t)$ and the principal eigenvalue $\lambda_1(b_0) < 1$.
• Hypothesis (H2): An upper comparison at infinity involving a weight function $b_\infty(t)$ and the principal eigenvalue $\mu_1(b_\infty) > 1$.

The main theorem (Theorem 3.1) proves that under these hypotheses, the fixed point operator $A$ (derived from the auxiliary equation) has a fixed point in the cone $K$. This fixed point corresponds to a positive solution of the original Minkowski-curvature equation with $\|u'\|_\infty < 1$.

Furthermore, the author derives Corollary 4.1, a directly verifiable asymptotic criterion. This result allows the comparison weights $b_0$ and $b_\infty$ to be determined directly from the uniform asymptotic slopes of $f(t, s)$ as $s \to 0^+$ and $s \to +\infty$, respectively, without requiring abstract construction of the weights.

Significance and Claims
The paper claims to fill a gap in the literature regarding mixed boundary conditions (Neumann–Dirichlet) combined with general, non-separable, sign-changing nonlinearities. Unlike previous works that relied on nonnegative assumptions or specific structural forms (e.g., $a(t)g(u)$), this work handles a general continuous $f(t, u)$.

The significance lies in the specific mechanism used to handle the sign-changing nature:

• The negative part of the nonlinearity is controlled explicitly by a linear bound rather than abstract cone invariance.
• The combination of a linear shift, Green's kernel, and principal-eigenvalue comparisons yields a robust framework for cone expansion near zero and compression at infinity.
• The paper provides a concrete example (Example 5.1) demonstrating that the class of sign-changing functions covered by the theorem is nonempty, verifying all assumptions explicitly for a specific trigonometric nonlinearity.

The author maintains a modest scope, focusing strictly on the existence of positive solutions and the derivation of verifiable criteria, rather than attempting a classification of global bifurcation curves or multiplicity results.