The convergence of nonnegative solutions for the family of problems − Δ<sub>p</sub>u = λe<sup>u</sup> as p →∞

Mihăilescu, Mihai; Stancu−Dumitru, Denisa; Varga, Csaba

doi:10.1051/cocv/2017048

Geodesic

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The convergence of nonnegative solutions for the family of problems − Δ_pu = λe^u as p →∞

Mihăilescu, Mihai ¹ ; Stancu−Dumitru, Denisa ¹ ; Varga, Csaba ¹

ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 2, pp. 569-578

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Résumé

Let $Ω \subset ℝ^{n}, (N \geq 2)$ be a bounded domain with smooth boundary. We show the existence of a positive real number $λ^{*}$ such that for each $λ \in (0, λ^{*})$ 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 $p \to \infty$ and we show that it converges uniformly to the distance function to the boundary of the domain.

MR Zbl

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

Affiliations des auteurs :

Mihăilescu, Mihai ¹ ; Stancu−Dumitru, Denisa ¹ ; Varga, Csaba ¹

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     author = {Mih\u{a}ilescu, Mihai and Stancu\ensuremath{-}Dumitru, Denisa and Varga, Csaba},
     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},
     volume = {24},
     number = {2},
     year = {2018},
     doi = {10.1051/cocv/2017048},
     zbl = {1404.35311},
     mrnumber = {3816405},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2017048/}
}

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Mihăilescu, Mihai; Stancu−Dumitru, Denisa; Varga, Csaba. The convergence of nonnegative solutions for the family of problems − Δ_pu = λe^u as p →∞. ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 2, pp. 569-578. doi: 10.1051/cocv/2017048

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