Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
7222538 | Nonlinear Analysis: Theory, Methods & Applications | 2018 | 16 Pages |
Abstract
Let Ω=R2âB(0,1)¯ be the exterior of the closed unit ball. We prove the existence of extremal constant-sign solutions in D01,2(Ω) of the following logistic-type boundary value problem âÎu=a(x)(λuâg(u))inΩ,u=0onâΩ=âB(0,1),under the assumption that the nonnegative coefficient a decays like |x|â2âα with α>0, and the growth for the continuous nonlinearity g:RâR at zero and at infinity is superlinear which includes even exponential growth. We are looking for solutions in the space D01,2(Ω) which is the completion of Ccâ(Ω) with respect to the âââ
â2,Ω-norm. For general unbounded domains in R2 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 RN with Nâ¥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 D01,2(Ω).
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Authors
Siegfried Carl, David G. Costa, Hossein Tehrani,