A priori estimates and multiplicity for systems of elliptic PDE with\n natural gradient growth

2019/10/07 by Gabrielle Nornberg, Nornberg, Gabrielle, Delia Schiera +3
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1910.03083

openalex publication_date 2019/10/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider fully nonlinear uniformly elliptic cooperative systems with\nquadratic growth in the gradient, such as -F_i(x, u_i, Du_i, D² u_i)-\n
langle M_i(x)D u_i, D u_i
rangle =
lambda c_i1(x) u₁ +
cdots +
lambda\nc_in(x) u_n +h_i(x), for i=1,\⋯,n, in a bounded C^1,1 domain\n\Ω\⊂ \ℝ^N with Dirichlet boundary conditions; here n\≥\n1, \λ \∈\ℝ, c_ij, , h_i \∈ L^\∞(\Ω), c_ij\≥\n0, M_i satisfies 0<\μ₁ I\≤ M_i\≤ \μ₂ I, and F_i is an uniformly\nelliptic Isaacs operator.\n We obtain uniform a priori bounds for systems, under a weak coupling\nhypothesis that seems to be optimal. As an application, we also establish\nexistence and multiplicity results for these systems, including a branch of\nsolutions which is new even in the scalar case.\n

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