Large solutions to the p-Laplacian for large p

García-Melián, Jorge; Rossi, Julio; Lis, José
February 2008
Calculus of Variations & Partial Differential Equations;Feb2008, Vol. 31 Issue 2, p187
Academic Journal
In this work we consider the behaviour for large values of p of the unique positive weak solution u p to Δ p u = u q in Ω, u = +∞ on $$\partial\Omega$$ , where q > p − 1. We take q = q( p) and analyze the limit of u p as p → ∞. We find that when q( p)/ p → Q the behaviour strongly depends on Q. If 1 < Q < ∞ then solutions converge uniformly in compacts to a viscosity solution of $${\rm max}\{- \Delta_\infty{u}, -|\nabla u| +u^Q \} = 0$$ with u = +∞ on $$\partial\Omega$$ . If Q = 1 then solutions go to ∞ in the whole Ω and when Q = ∞ solutions converge to 1 uniformly in compact subsets of Ω, hence the boundary blow-up is lost in the limit.


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