On a necessary condition for removing singularities of solutions of nonlinear elliptic inequalities

Technical Summary: On a Necessary Condition for Removing Singularities of Solutions of Nonlinear Elliptic Inequalities

Problem Statement
The paper investigates the removability of singularities for solutions of nonlinear elliptic differential inequalities of the form:
$$\Delta^{m/2} u \geq f(x)g(u)$$
defined in the punctured unit ball $B_1 \setminus \{0\} \subset \mathbb{R}^n$ ($n \geq 2$), where $m \geq 2$ is an even integer. The functions $f$ and $g$ satisfy specific regularity and growth conditions: $f \in L^\infty_{loc}(B_1 \setminus \{0\})$ and $g: [0, \infty) \to [0, \infty)$ is non-decreasing with $g(0)=0$ and $g(\zeta) > 0$ for $\zeta > 0$.

A singularity at the origin is defined as removable if the solution $u$ and the term $f(x)g(|u|)$ belong to $L^1(B_1)$, allowing the inequality to hold in the distributional sense over the entire ball $B_1$. The central objective is to establish a necessary condition on the nonlinearity $g$ such that any solution of the inequality possesses a removable singularity at zero.

Methodology
The authors employ a constructive approach to prove that if a specific integral condition on $g$ fails, a non-removable singularity must exist. The methodology proceeds as follows:

1. Reduction to Radial ODEs: The problem is reduced from the partial differential inequality to a radial ordinary differential inequality. By utilizing the properties of the Laplacian in polar coordinates, the authors construct a radial solution $u(x) = w(|x|)$ that satisfies the inequality if $w$ satisfies a related radial problem involving the operator $\left(\frac{1}{r^{n-1}}\left(r^{n-1}\frac{d}{dr}\right)\right)^{m/2}$.
2. Fixed Point and Iterative Construction: The proof relies on Theorem 2.2, which asserts the existence of a positive, decreasing solution to a specific radial inequality on the interval $(0, 1]$. This is achieved through:
   • Defining a sequence of integral operators $A_i$ acting on continuous functions.
   • Establishing a priori estimates and inequalities (e.g., equations 2.4–2.11) to control the growth and decay of these operators.
   • Utilizing the Bohl–Brouwer fixed point theorem to find a fixed point $w_0$ on a sub-interval $[r_0, 1]$.
   • Extending this solution to $(0, r_0]$ by solving a Cauchy problem involving the inverse of the nonlinearity $g^{-1/m}$.
3. Divergence Argument: The core of the argument hinges on the behavior of the integral $\int_1^\infty g^{-1/m}(\zeta)\zeta^{1/m-1} d\zeta$. If this integral diverges, the constructed solution $w(r)$ grows sufficiently fast as $r \to 0$ such that $u(x) \notin L^1(B_1)$, thereby proving the singularity is non-removable.

Key Contributions and Results
The primary result is Theorem 2.1, which states:
If $g: (0, \infty) \to (0, \infty)$ is a non-decreasing continuous function satisfying the divergence condition:
$$\int_1^\infty g^{-1/m}(\zeta)\zeta^{1/m-1} d\zeta = \infty,$$
then the inequality $\Delta^{m/2} u \geq f(x)g(u)$ admits a positive solution with a non-removable singularity at zero for any $f \in L^\infty_{loc}(B_1 \setminus \{0\})$.

Consequently, the condition:
$$\int_1^\infty g^{-1/m}(\zeta)\zeta^{1/m-1} d\zeta < \infty$$
is established as a necessary condition for the removability of singularities for any solution of the inequality.

The paper contextualizes this result by relating it to known conditions:

• For $m=2$, the condition is equivalent to the well-known Keller–Osserman blow-up condition.
• The result generalizes previous findings for power-type nonlinearities (e.g., $\Delta u \geq |x|^s u^\lambda$) and logarithmic nonlinearities, recovering exact critical exponents such as the Brezis–Véron condition.

Significance and Claims
The authors claim that while the integral condition (1.3) (or its generalization for $m>2$) is not sufficient to guarantee removability (as it does not account for the behavior of $f$ near zero or potential growth restrictions in linear cases), it is strictly necessary.

The paper asserts that if this integral condition is violated, no matter how "small" or well-behaved the coefficient function $f$ is (provided it is locally bounded away from zero), one can always construct a solution that blows up at the origin in a way that prevents the singularity from being removable. This provides a sharp boundary for the class of nonlinearities $g$ that allow for the removal of singularities in the context of higher-order nonlinear elliptic inequalities. The work clarifies the exact threshold between removable and non-removable singularities, independent of the specific choice of the forcing term $f$.