SOME MAXIMUM PRINCIPLES FOR SOLUTIONS OF A CLASS OF PARTIAL DIFFERENTIAL EQUATIONS IN Ω ⊂Rn

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.


The maximum principles.
The new results are in the following two theorems.
where α is a positive constant, and f (u) is a positive, nondecreasing, differentiable function; and if ∆u = 0 on ∂Ω, then u attains its maximum on ∂Ω.
Proof.Let u ∈ C 4 (Ω)∩C 2 ( Ω) be a solution of (2.1), then u satisfies the equations Now define Denoting one of the variables, x k , by β and differentiating (2.3) twice with respect to β, we get (2.4) If we sum over all β = x k , we get 2) into (2.5),we get Since f (u) ≥ 0 we see that ∆L ≥ 0. But ∆L ≠ 0 as u is nonconstant.Hence L is a nonconstant subharmonic function.And it follows from the maximum principle of subharmonic functions that L(x) cannot attain its maximum at any interior point of Ω, that is, for some x 0 ∈ ∂Ω and for all x ∈ Ω.
(2.8)However, since (∆u(x 0 )) = 0, it yields 2α or, since α > 0, we have (2.10) Theorem 2.2.Let u be a nonconstant solution of the partial differential equation where α is a nonpositive constant, k is an odd integer and g > 0 is twice continuously differential and is such that (2.12)

.13)
For some x 0 ∈ ∂Ω and for all x ∈ ∂Ω provided ∆u = 0 on ∂Ω, g x 0 < g(x). (2.14) Proof.A nonconstant solution u of (2.11) satisfies the equations As in the proof of Theorem 2.1, we consider the function (2.16) Differentiating twice with respect to x k and summing over all x k ,s, we get (2.17) Since α ≤ 0 and g > 0, we can conclude with the help of (2.12) and (2.15) that L is subharmonic and it follows from the maximum principles of subharmonic functions that there exist a point x 0 ∈ ∂Ω such that for all x ∈ Ω.But since ∆u = 0 on ∂Ω, the assertion is proved with the help of (2.14).where α ≤ 0, β > 0 are constants and f (u) is a positive, nondecreasing, and a differentiable function.
(d) One may give extensions of the maximum principle to solutions of equations as ∆(h∆u) + α(h∆u) k + gu = 0 under suitable assumptions.