A limiting problem for local/non-local p-Laplacians with concave–convex nonlinearities

Abstract

In this manuscript, we deal with an equation involving a combination of quasi-linear elliptic operators of local and non-local nature with p-structure, and concave?convex nonlinearities. The prototypical model is given by {-Δ_pu+(-)_p^su=λ_pu^q(x)+ur(x)inΩ, u(x)>0inΩ, u(x)=0onR^n\Ω, where Ω ⊂ R^n is a bounded and smooth domain, s∈ (0 , 1) , 2 ≤ p< ∞, 0 < q(p) < p- 1 < r(p) < ∞ and 0 < λp< ∞, being Δ p and (-Δ)ps the p-Laplace and fractional p-Laplace operators, respectively. We study existence and global uniform and explicit boundedness results to weak solutions. Then, we perform an asymptotic analysis for the limit of a family of weak solutions {u_p}_p≥2^ as p→ ∞, which converges, up to a subsequence (under suitable assumptions on the problem data), to a non-trivial profile with uniform and explicit bounds, enjoying of a universal Lipschitz modulus of continuity, and verifying a nonlinear limiting PDE in the viscosity sense, which exhibits both local/non-local character.