Maximum Principles for Fourth Order Nonlinear Elliptic Equations with Applications

The paper is devoted to prove maximum principles for a certain functionals defined for solution of the fourth order nonlinear elliptic equation. The maximum principle so obtained is used to prove the nonexistence of nontrivial solutions of the fourth order nonlinear elliptic equation with some zero boundary conditions. Hopf’s maximum principle is the main ingredient.


Introduction
There are large number of papers related to study of existence of solution of second order, nonlinear elliptic boundary value problems.It is well known that maximum principle and comparison principle are the two sides of the same coin.It is popular that maximum principle, comparison principle and existence results go hand in hand (cf.[1], [2], [4]).
It is well known that there is no maximum principle for fourth order elliptic partial differential equations comparable to the maximum principle for second order elliptic equations.In [5] Miranda obtained the first such result for biharmonic equation, ∆ 2 u = 0, by proving that the functional is subharmonic on its domain.Since, then many researchers have employed this technique on various class of fourth order partial differential equations.In [7], for example, Schaefer utilized auxiliary functions of type (1.1) to study semilinear equations of the form in a plane domain.Recently Dhaigude and Gosavi [3] extended a maximum principle due to Schaefer [7] for a class of fourth order semilinear elliptic equations to a more general fourth order semilinear elliptic equation of the form In this paper, we study the existence problem for fourth order nonlinear elliptic equation of the form For simplicity, we use the summation convention and denote partial derivatives ∂u ∂x i by u ,i and This paper is organized as follows.In section 2 we develop a maximum principle for a class of fourth order nonlinear elliptic equations.The maximum principle will be used to deduce the non-existence of non-trivial solutions of the boundary value problem under consideration in the last section.

Maximum Principle
Suppose Ω is a plane domain bounded with smooth boundary ∂Ω.The following Lemma [8] is useful to prove our results.
Lemma 2.1.For a sufficiently smooth function v the inequality Now, we prove the following maximum principles for the function P denoted by P = |∇u(x)| 2 − u∆u, which will be the main result of this paper.
Fourth Order Nonlinear Elliptic Equations 11 Proof: By straightforward computations By maximum principle [6], the result follows.
Thus, we claim that our results are more general.) is bounded and ∆b ≤ 0. Then the function assume its nonnegative maximum on ∂Ω unless P < 0 in Ω.
Proof: By straightforward computations The following Lemma is useful in proving the non-existence result in the next section of the paper.Lemma 2.9.[7] If (2.2) is satisfied and if u is a C 4 solution of (2.1) which vanishes on ∂Ω, then where |∇u| 2 M = max|∇u| 2 and A is the area of Ω.
Proof: From Theorem 2.2 we know that P attains its maximum on the boundary.Also we know that u = 0 on ∂Ω.Let |∇u| 2 M = max|∇u| 2 on ∂Ω.We have

Applications
In this section as an application of our maximum principle we prove nonexistence of nontrivial solutions u(x) of the following boundary value problem and Consequently |∇u| = 0 in Ω and by continuity u ≡ 0 in Ω ∪ ∂Ω.This is a contradiction.Hence there is no nontrivial solution of (3.(3.12) Since Ω is convex, k > 0. So ∂P ∂n (Q) > 0 is impossible.Therefore, in this case no nontrivial solution exists.
Case II.Suppose P is a constant say c in Ω ∪ ∂Ω.Then we have For a bounded convex domain with a continuously turning tangent on the boundary, k = 0. Moreover c = 0, for, if c = 0, then |∇u| M = 0 and by Lemma 2.9 and reasoning as in Theorem 2.2 we are led to that u ≡ 0 in Ω.Thus P = c is impossible.As neither case is possible, we conclude that no nontrivial solution exists.
1) -(3.2).It is by contradiction.Assume on the contrary that a nontrivial solution u of the given BVP (3.3) -(3.4) exists.We have P as defined in (2.3).Then by Theorem 2.2, P takes its maximum on the boundary ∂Ω at a point, say Q.By Hopf's second maximum principle, either ∂P ∂n (Q) > 0 or P is constant in Ω ∪ ∂Ω.Case I. Suppose ∂P ∂n (Q) > 0 holds.Differentiate P partially in the normal direction and use boundary condition (3.4) to get 2.