Some maximum principles for solutions of a class of partial differential equations inΩ⊂ℝn

Mohammad Almahameed

doi:10.1155/s0161171200004877

Abstract

We find maximum principles for solutions of semilinear elliptic partial differential equations of the forms: (1)Δ2u+αf(u)=0,α∈ℝ+and (2)ΔΔu+α(Δu)k+gu=0,α≤0in some regionΩ⊂ℝn.

Highlights

In [1], Chow and Dunninger proved the following result: let u ∈ C4(Ω) ∩ C2(Ω ) be a nonconstant solution of∆2u + αu = 0 in ΩCRn, α ∈ R+, ∆u = 0 on ∂Ω. (1.1) u satisfies a maximum principle
We find maximum principles for solutions of semilinear elliptic partial differential equations of the forms: (1) ∆2u+αf (u) = 0, α ∈ R+ and (2) ∆∆u + α(∆u)k + gu = 0, α ≤ 0 in some region Ω ⊂ Rn
We extend this result to solutions of semilinear partial differential equations of the forms: (1) ∆2u + αf (u) = 0, where α is a positive constant and f (u) is a positive, nondecreasing, differentiable function

Summary

Introduction

We find maximum principles for solutions of semilinear elliptic partial differential equations of the forms: (1) ∆2u+αf (u) = 0, α ∈ R+ and (2) ∆∆u + α(∆u)k + gu = 0, α ≤ 0 in some region Ω ⊂ Rn. Keywords and phrases. In [1], Chow and Dunninger proved the following result: let u ∈ C4(Ω) ∩ C2(Ω ) be a nonconstant solution of We extend this result to solutions of semilinear partial differential equations of the forms: (1) ∆2u + αf (u) = 0, where α is a positive constant and f (u) is a positive, nondecreasing, differentiable function.

Results

Conclusion