Boundary blow-up solutions of pp-Laplacian elliptic equations with a weakly superlinear nonlinearity

This paper studies the asymptotic behavior near the boundary for boundary blow-up solutions to pp-Laplacian elliptic equations −Δpu=a(x)h(u)−b(x)f(u)−Δpu=a(x)h(u)−b(x)f(u) in a smooth bounded domain ΩΩ of RNRN with N≥2N≥2, where Δpu=div(∣∇u∣p−2∇u) with p>1p>1. We assume that f(u)f(u) does not grow like uquq with q>p−1q>p−1 or faster at infinity, but behaves like up−1lnαuup−1lnαu as u→∞u→∞ for some α>pα>p, and h(u)h(u) does not grow faster than up−1up−1 at infinity. This case is more difficult to handle than those two cases (f(u)f(u) grows like uquq with q>p−1q>p−1 or faster at infinity). Under suitable conditions on the weight function b(x)b(x), which may vanish on ∂Ω∂Ω, we obtain the first order expansion of the large solutions near the boundary. Uniqueness of such solutions would then follow as a consequence.