A bifurcation problem governed by the boundary condition I

Abstract

We deal with positive solutions of ∆u = a(x)u p in a bounded smooth domain Ω ⊂ R N subject to the boundary condition ∂u/∂ν = λu, λ a parameter, p > 1. We prove that this problem has a unique positive solution if and only if 0 < λ < σ1 where, roughly speaking, σ1 is finite if and only if |∂Ω ∩ {a = 0}| > 0 and coincides with the first eigenvalue of an associated eigenvalue problem. Moreover, we find the limit profile of the solution as λ → σ1.

Positive solutions for some indefinite nonlinear eigenvalue elliptic problems with Robin boundary conditions

“…In fact, (H M ± ) is assumed to avoid regularity issues, see [12]. In Figure 2 we have represented two different admissible domains.…”

Nonnegative solutions to an elliptic problem with nonlinear absorption and a nonlinear incoming flux on the boundary

“…Finally, the case r = 1, λ = 0, both for p > 1 and p < 1, and with a parameter in the boundary condition, is considered in [20] and [21].…”

An elliptic problem with an indefinite nonlinearity and a parameter in the boundary condition

“…In the next step we follow argument of Proposition 5 in [14] (page 506). We consider the auxiliary problem −Δv + Mv = f (x) in Ω, ∂v ∂ν = λv on ∂Ω, (7.8) where f (x) = −a(x)u p + Mu and M > 0 is a constant.…”