Existence and nonexistence of nontrivial solutions for a critical
biharmonic equations under the Steklov boundary conditions
Abstract
In this paper, we study the existence and nonexistence of nontrivial
solutions to the following critical biharmonic problem with the Steklov
boundary conditions
Δ2=+Δ+||2**-2 in ,
=Δ+=0 on , where ,, ∈ , ⊂ N( ≥ 5) is a
unit ball, 2** = 2/N-4 denotes the critical Sobolev
exponent for the embedding 2() →2**
() and is the outer normal derivative of on . Under
some assumptions on , and , we prove the existence of nontrivial
solutions to the above biharmonic problem by the Mountain pass theorem
and show the nonexistence of nontrivial solutions to it by the Pohozaev
identity.