Limits as of -Laplacian concave–convex problems
Abstract We study the behavior as p → ∞ of the sequel of positive weak solutions of the concave–convex problem (P ) { − div ( | ∇ u | p − 2 ∇ u ) = λ u q ( p ) + u r ( p ) in Ω u > 0 in Ω u = 0 on ∂ Ω , where Ω ⊂ R n is a bounded domain, λ > 0 and the exponents q , r satisfy lim p → ∞ q ( p ) p − 1 = Q , lim p → ∞ r ( p ) p − 1 = R , with 0 Q 1 R . We characterize any positive uniform limit of a sequence of weak solutions of (P) as a viscosity solution of min { | ∇ u Λ | − max { Λ u Λ Q , u Λ R } , − Δ ∞ u Λ } = 0 in Ω . Notice that the limit process decouples the nonlinearity. We obtain existence, non-existence and global multiplicity of positive viscosity solutions of the limit problem in terms of the parameter Λ .