Abstract
We discuss the asymptotic behavior of positive solutions of the quasilinear elliptic problem −Δpu = a up−1−b(x) uq, u|∂ Ω = 0, as q → p − 1 + 0 and as q → ∞, via a scale argument. Here Δp is the p-Laplacian with 1 < p ∞ and q > p − 1. If p = 2, such problems arise in population dynamics. Our main results generalize the results for p = 2, but some technical difficulties arising from the nonlinear degenerate operator −Δp are successfully overcome. As a by-product, we can solve a free boundary problem for a nonlinear p-Laplacian equation.
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