Extremal solutions of logistic-type equations in exterior domain in R2. (November 2018) - Carl, Siegfried; Costa, David G.; Tehrani, Hossein
Abstract: Let Ω = R 2 ∖ B ( 0, 1 ) ¯ be the exterior of the closed unit ball. We prove the existence of extremal constant-sign solutions in D 0 1, 2 ( Ω ) of the following logistic-type boundary value problem − Δ u = a ( x ) ( λ u − g ( u ) ) in Ω, u = 0 on ∂ Ω = ∂ B ( 0, 1 ), under the assumption that the nonnegative coefficient a decays like | x | − 2 − α with α > 0, and the growth for the continuous nonlinearity g : R → R at zero and at infinity is superlinear which includes even exponential growth. We are looking for solutions in the space D 0 1, 2 ( Ω ) which is the completion of C c ∞ ( Ω ) with respect to the ‖ ∇ ⋅ ‖ 2, Ω -norm. For general unbounded domains in R 2 including the whole plane, this completion may result in objects that do not belong to any function space. This is one of the main reasons to consider the problem in the exterior of the unit ball instead in the whole plane. Unlike in the situation of R N with N ≥ 3, i.e. N > p = 2, the behavior of the solutions in the case N = p = 2 considered here is significantly different, in particular, it will be seen that the solutions are not decaying to zero at infinity, and instead are bounded away from zero. To prove our main results, new tools have to be developed here such as for example a Hopf-type lemma and a sub–supersolution method in D 0 1, 2 ( Ω ) .