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Positive radial solutions of a nonlinear boundary value problem

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  • In this work we study the following quasilinear elliptic equation:

    $\left\{ {\begin{array}{*{20}{l}}{ - {\rm{div}}(\frac{{|x{|^\alpha }\nabla u}}{{{{(a(|x|) + g(u))}^\gamma }}}) = |x{|^\beta }{u^p}}&{{\rm{in}} \ \Omega }\\{u = 0}&{{\rm{on}}\;\;\;\;\partial \Omega }\end{array}} \right.$

    where $ a $ is a positive continuous function, $ g $ is a nonnegative and nondecreasing continuous function, $ Ω = B_R $, is the ball of radius $ R>0 $ centered at the origin in $ \mathbb{R} ^N $, $N≥3 $ and, the constants $ α,β∈\mathbb{R} $, $ γ∈(0,1) $ and $ p>1 $.

    We derive a new Liouville type result for a kind of "broken equation". This result together with blow-up techniques, a priori estimates and a fixed-point result of Krasnosel'skii, allow us to ensure the existence of a positive radial solution. In this paper we also obtain a non-existence result, proven through a variation of the Pohozaev identity.

    Mathematics Subject Classification: 35B09, 35B45, 35B53, 35J62, 58J20.


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