Maximum Principles for Fourth Order Nonlinear Elliptic Equations with Applications

Gajanan Chandrakantrao Lomte

doi:10.5269/bspm.v32i1.17305

Abstract

The paper is devoted to prove maximum principles for a certain func- tionals defined for solution of the fourth order nonlinear elliptic equation. The max- imum 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.

Highlights

Then the functionP = |∇u(x)|2 − u∆u assume its maximum on ∂Ω
We study the existence problem for fourth order nonlinear elliptic equation of the form
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

Summary

Then the function

Assume its nonnegative maximum on ∂Ω unless P < 0 in Ω. ∇Pb + ∆bPb = 2u,iku,ik − (∆u)2 − u∆2u. It follows from Lemma 2.1 and assumptions (2.8) that Pb satisfies b∆Pb − 2b2∇. If a = 0 in (2.7) the maximum principle in Sperb ( [8], Theorem 10.2 ) follows. Proof: From Theorem 2.2 we know that P attains its maximum on the boundary. As an application of our maximum principle we prove nonexistence of nontrivial solutions u(x) of the following boundary value problem. Theorem 2.2 and boundary condition (3.2) gives u,iu,i − u∆u ≤ 0. If (2.2) is satisfied in a convex domain Ω no nontrivial solution of (3.3) - (3.4) exists. By Theorem 2.2, P takes its maximum on the boundary ∂Ω at a point, say Q.

But from we have