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One-critical-exponent regime with positive coefficients

Determine the existence or nonexistence of weak solutions u in the weighted Sobolev space D^{1,2}_ρ(R^N_+) to the boundary value problem −div(ρ(x_N)∇u) = a|u|^{p−2}u in R^N_+ and −∂u/∂x_N = b|u|^{q−2}u on R^{N−1}, where a>0, b>0, ρ satisfies the paper’s standing hypotheses on positive weights, and exactly one exponent is critical, namely either p = 2N/(N−2) with q ∈ (1, 2(N−1)/(N−2)) or p ∈ (1, 2N/(N−2)) with q = 2(N−1)/(N−2).

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Background

The paper proves existence of positive weak solutions for the weighted half-space problem when both exponents are subcritical and a,b>0, and establishes nonexistence when both exponents are critical. It also proves nonexistence for certain sign combinations (a>0,b≤0 with p critical; a≤0,b>0 with q critical).

However, when both coefficients are positive and exactly one exponent is critical (either p=2* with q subcritical or q=2_* with p subcritical), the analysis is not settled by the results provided. The authors highlight this specific regime as an open case, distinguishing it from the treated subcritical and fully critical regimes.

References

Another open case arises when $a, b > 0$ and either $p$ or $q$ reaches the critical exponent; that is, when $p = 2\ast$ and $q \in (1, 2_\ast)$, or $p \in (1, 2\ast)$ and $q = 2_\ast$.

— Existence and Nonexistence Breaking Results For a Weighted Elliptic Problem in Half-Space (2510.05999 - Ó et al., 7 Oct 2025) in Introduction and main results, end of subsection 'Pohozaev identity and nonexistence results' (bullet list following Theorem Nonexistence2)