The Boundary Trace of Positive Solutions of Semilinear Elliptic Equations: the Subcritical Case

Abstract

We study the generalized boundary value problem for nonnegative solutions of of −∆u + g(u) = 0 in a bounded Lipschitz domain Ω, when g is continuous and nondecreasing. Using the harmonic measure of Ω, we define a trace in the class of outer regular Borel measures. We amphasize the case where g(u) = |u| q−1 u, q > 1. When Ω is (locally) a cone with vertex y, we prove sharp results of removability and characterization of singular behavior. In the general case, assuming that Ω possesses a tangent cone at every boundary point and q is subcritical, we prove an existence and uniqueness result for positive solutions with arbitrary boundary trace.1991 Mathematics Subject Classification. 35K60; 31A20; 31C15; 44A25; 46E35.

Asymptotic behavior of solutions to the σ k -Yamabe equation near isolated singularities

“…This result was later generalized by D. Finn [16] to the case of ∂Ω consisting of smooth submanifolds of dimension > (n − 2)/2 and with boundary. For more recent development related to the negative scalar curvature case, see [33], [34], [40] and the references therein. The consideration of singular solutions of equations of type (8) can be considered as a natural generalization of these known results.…”

On Uniqueness of Large Solutions of Nonlinear Parabolic Equations in Nonsmooth Domains

“…The conclusion of the proof is contradiction, following an idea introduced in [8] and developped by [12] in the elliptic case. We assume u = u 0 , thus u < u 0 .…”

The parabolic logistic equation with blow-up initial and boundary values

“…Combining (3.6) and (3.8), and taking into account that > 0 can be arbitrarily small, we deduce [(p − 1)b(x 0 , 0)t] 1 p−1 u(x 0 , t) → 1 as t → 0, which is the desired result. Finally, we use the convex function technique introduced by Marcus and Véron [9,10] to show the uniqueness of positive solutions of (1.1) for the case β = 0.…”

“…The equations in [1, 2, 11] also only involve constant coefficients. Theorem 1.4 follows from Theorem 1.1 and a convex function technique of Marcus and Véron [9,10].…”