A uniqueness result for a semipositone p-Laplacian problem on the exterior of a ball

作者：Shivaji R; Sim Inbo^*; Son Byungjae

来源：Journal of Mathematical Analysis and Applications, 2017, 445(1): 459-475.

DOI：10.1016/j.jmaa.2016.07.029

• Uniqueness results
• Semipositone problems
• p-Laplacian
• Exterior domains
• Positive radial solutions

We consider steady state reaction diffusion equations on the exterior of a ball, namely, boundary value problems of the form:
{-Delta(p)u = lambda K(vertical bar x vertical bar)f(u) in Omega(E),
u = 0 on vertical bar x vertical bar = r(0),
u -> 0 when vertical bar x vertical bar -> infinity,
where Delta(p)z := div(vertical bar del z vertical bar(p-2)del z), 1 < p < n, lambda is a positive parameter, r(0) > 0 and Omega(E) := {x is an element of > R-n vertical bar vertical bar x vertical bar > r(0)}. Here the weight function K is an element of C-1[r(0), infinity) satisfies K(r) > 0 for r >= r(0), lim(r ->infinity) K(r) = 0, and the reaction term f is an element of C[0, infinity) boolean AND C-1(0, infinity) is strictly increasing and satisfies f(0) < 0 (semipositone), limsup(s -> 0+) sf'(s) < infinity, lim(s ->infinity)f(s) = infinity, lim(s ->infinity) f(s)/s(p-1) = 0 and f(s)/s(q) is nonincreasing on [a, infinity) for some a > 0 and q is an element of (0, p - 1). For a class of such steady state equations it turns out that every nonnegative radial solution is strictly positive in the exterior of a ball, and exists for lambda >> 1. We establish the uniqueness of this positive radial solution for lambda >> 1.

• 出版日期2017-1-1