临界半线性双调和方程非平凡解的存在性

Existence of nontrivial solutions for critical semilinear biharmonic equations

在有界光滑区域ΩR^N上研究临界半线性双调和方程Δ~2u=λu+|u|^(q-2)u,λ>0,u∈H_0~1(Ω)∩H^2(Ω)非平凡解的存在性.利用极小极大原理和山路引理,证明方程所对应的泛函存在临界点,从而得到方程至少存在一个非平凡解的结论.

The existence of nontrivial solutions for the problem Δ~2u =λu＋｜u｜^（q-2）u,λ0,u H_0~1（Ω）∩ H^2（Ω） is discussed in this paper under the condition that ΩR^N is a bounded smooth domain.Applying the mini-max principle and mountain-pass lemma,a critical point of the corresponding functional of the equation is obtained,indicating the existence of nontrivial weak solutions.