Nontrivial convex solutions on a parameter of impulsive differential equation with Monge-Amp\xe8re operator

Abstract

The authors consider the impulsive differential equation with Monge-Ampère operator in the form of \t\t\t{((u′(t))n)′=λntn−1f(−u(t)),t∈(0,1),t≠tk,k=1,2,…,m,Δ(u′)n|t=tk=λIk(−u(tk)),k=1,2,…,m,u′(0)=0,u(1)=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\textstyle\\begin{cases} ( (u'(t) )^{n} )'=\\lambda nt^{n-1}f (-u(t) ), \\quad t\\in(0,1), t\\neq t_{k}, k=1, 2, \\ldots, m, \\\\ \\Delta (u' )^{n}|_{t=t_{k}}=\\lambda I_{k} (-u(t_{k}) ), \\quad k=1, 2, \\ldots , m, \\\\ u'(0)=0, \\quad\\quad u(1)=0, \\end{cases} $$\\end{document} where λ is a nonnegative parameter and ngeq1. We show the existence, uniqueness, and continuity results. Our approach is largely based on the eigenvalue theory and the theory of α-concave operators. The nonexistence result of a nontrivial convex solution is also studied by taking advantage of the internal geometric properties related to the problem.