Sharpness of the integral-method threshold below the critical dimension

Determine whether the condition q>(p-2)^2/4 required by the integral method for the p-Laplacian equation Δ_p u=u^{-q} in dimensions 2≤N<p is sharp.

Background

The paper studies positive solutions of negative-power quasilinear equations, including the p-Laplacian equation Δ_p u=u{-q}, and establishes nonexistence results for stable solutions in several dimensional and exponent ranges. The authors identify a threshold q>(p-2)2/4 in the range 2≤N<p arising from the condition α₀<1−p required by their integral estimates.

For dimensions N≥p, the paper proves sharp nonexistence results by constructing stable solutions at or below the relevant critical thresholds. However, in the lower-dimensional range 2≤N<p, the authors do not determine whether the integral-method condition q>(p-2)2/4 is optimal, leaving open the question of sharpness of this threshold for the p-Laplacian negative-power equation.

References

When 2≤ N<p, the integral method needs q>\frac{(p-2)2}4, which is exactly the condition \alpha_0<1-p for the p-Laplacian, and we do not know whether this is sharp.

— Solutions stable outside a compact set for anisotropic quasilinear equations with negative exponent or exponential nonlinearity  (2610.06326 - Huynh et al., 5 Oct 2026) in Section 1, Introduction