A limiting problem for local/non-local \(p\)-Laplacians with concave-convex nonlinearities
The authors 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. Such a prototypical model is given by \[ \begin{cases} \mathcal {P}_{s,p}u(x)=\lambda_pu^q(x)+u^r(x)&\mbox{in}~~x\in\Omega,\\ u(x)>0&\mbox{in}~~\Omega,\\ u(x)=0&\mbox{on}~~\mathbb R^N\backslash\Omega, \end{cases}\tag{1.1} \] where \(\Omega\subset\mathbb R^n\) is a bounded and smooth domain, \(\mathcal {P}_{s,p}u(x)=-\Delta_p u(x)+(-\Delta)^s_pu(x)\), \(2 \leq p < \infty\), and the exponents \(q\) and \(r\) are assumed to satisfy \(0 < q(p) < p-1 < r(p) < \infty\) and \(\lambda_p\in (0,\infty)\). The authors study existence and global uniform and explicit boundedness of the weak solutions of (1.1), by using the sub-solution analysis, the viscosity solution method and a weak comparison principle. Also, they study the asymptotic analysis for the limit of a family of weak solutions \(\{u_p\}_{p\geq2}\) as \(p\rightarrow\infty\), and show that \(u_p\rightarrow u_{\infty}\) uniformly on \(\overline{\Omega}\), up to a subsequence, 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.
Hölder \(\infty\)-Laplacian and \(\infty\)-Laplacian
scientific article; zbMATH DE number 7348016
scientific article