Non-homogeneous conormal derivative problem for quasilinear elliptic equations with Morrey data

A non-homogeneous conormal derivative problem is considered for quasilinear divergence form elliptic equations modeled on the $m$-Laplacian operator. The nonlinear terms are given by Carathéodory functions and satisfy controlled growth structure conditions with respect to the solution and its gradient, while their $x$-behaviour is controlled in terms of suitable Morrey spaces. Global essential boundedness is proved for the weak solutions, generalizing thus the classical $L^p$-result of Ladyzhenskaya and Ural’tseva to the framework of the Morrey scales.

💡 Research Summary

The paper studies a non‑homogeneous conormal derivative boundary value problem for quasilinear elliptic equations in divergence form that are modeled on the m‑Laplacian. The problem is set on a bounded Lipschitz domain Ω⊂ℝⁿ (n≥2) and reads

  div a(x,u,Du)=b(x,u,Du) in Ω,
  a(x,u,Du)·ν=ψ(x,u) on ∂Ω.

The vector field a, the lower‑order term b, and the boundary nonlinearity ψ are Carathéodory functions: measurable in x and continuous in (z,ξ). The authors impose a coercivity condition of order m, controlled growth bounds for a, b, and ψ with respect to the solution u and its gradient Du, and crucially they allow the x‑dependence of these coefficients to belong only to suitable Morrey spaces rather than classical Lebesgue spaces.

The main structural hypotheses are:

1. Coercivity: a(x,z,ξ)·ξ ≥ γ|ξ|^m − Λ|z|^{m*} − Λ φ₁(x)^{m/(m−1)} with φ₁∈L^{m}_{m−1}(Ω).

2. Controlled growth:
    |a| ≤ Λ(φ₁+|z|^{m*(m−1)/m}+|ξ|^{m−1}),
    |b| ≤ Λ(φ₂+|z|^{m*−1}+|ξ|^{m(m*−1)/m*}),
    |ψ| ≤ ψ₁+ψ₂|z|^{β},

where φ₂, ψ₁, ψ₂ belong to Morrey spaces L^{p_i}{λ_i}(Ω) and L^{q_i}{μ_i}(∂Ω) with parameters satisfying p₁>m/(m−1), p₂>mn/(mn−n+m), q₁>m(n−1)/