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 $$\begin{aligned} \left\{ \begin{array}{rclcl} -\Delta _p u + (-\Delta )^s_p u &{} = &{} \lambda _p u^q(x) + u^r(x) &{} \text{ in } &{} \Omega , \\ u(x)&{}>&{}0&{}\text{ in }&{} \Omega ,\\ u(x)&{} =&{} 0&{}\text { on } &{} \mathbb {R}^n {\setminus } \Omega , \end{array} \right. \end{aligned}$$ where $$\Omega \subset \mathbb {R}^n$$ is a bounded and smooth domain, $$s\in (0,1)$$ , $$2 \le p < \infty $$ , $$0<q(p)<p-1<r(p)<\infty $$ and $$0<\lambda _p< \infty $$ , being $$\Delta _p$$ and $$(-\Delta )_p^s$$ 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 $$\{u_p\}_{p\ge 2}$$ as $$p \rightarrow \infty $$ , 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.