The convergence of nonnegative solutions for the family of problems − Δpu = λeu as p →∞
ESAIM: Control, Optimisation and Calculus of Variations, Volume 24 (2018) no. 2, pp. 569-578.

Let Ω n , ( N 2 ) be a bounded domain with smooth boundary. We show the existence of a positive real number λ * such that for each λ ( 0 , λ * ) and each real number p > N the equation - Δ p u = λ e u in in Ω subject to the homogeneous Dirichlet boundary condition possesses a nonnegative solution u p . Next, we analyze the asymptotic behavior of u p as  p and we show that it converges uniformly to the distance function to the boundary of the domain.

DOI: 10.1051/cocv/2017048
Classification: 35D30, 35D40, 35J60, 47J30, 46E30
Keywords: Weak solutionviscosity solution, nonlinear elliptic equations, asymptotic behavior, distance function to the boundary
Mihăilescu, Mihai 1; Stancu−Dumitru, Denisa 1; Varga, Csaba 1

1
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     title = {The convergence of nonnegative solutions for the family of problems \ensuremath{-} {\ensuremath{\Delta}\protect\textsubscript{p}u} = \ensuremath{\lambda}e\protect\textsuperscript{u} as p {\textrightarrow}\ensuremath{\infty}},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {569--578},
     publisher = {EDP-Sciences},
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Mihăilescu, Mihai; Stancu−Dumitru, Denisa; Varga, Csaba. The convergence of nonnegative solutions for the family of problems − Δpu = λeu as p →∞. ESAIM: Control, Optimisation and Calculus of Variations, Volume 24 (2018) no. 2, pp. 569-578. doi : 10.1051/cocv/2017048. http://archive.numdam.org/articles/10.1051/cocv/2017048/

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