The existence of nontrivial solution for a class of sublinear biharmonic equations with steep potential well

摘要

In this paper, we study the following biharmonic equation: $$ extstyleegin{cases} Delta^{2}u - Delta u + lambda V( x)u = lpha( x) f(u) + mu K(x) ert u ert ^{q-2}u quadext{in } mathbb{R}^{N}, uin H^{2}(mathbb{R}^{N}), end{cases} $$ where (Delta^{2}u=Delta(Delta u)), (N4), (lambda0), (1 q2) and (muin[0,mu_{0}]). By using Ekelanda??s variational principle and Gigliardoa??Nirenberga??s inequality, we prove the existence of nontrivial solution for the above problem.KeywordsBiharmonic equation??Variational method??Steep potential well??MSC35J50??35J60??1 IntroductionIn this paper, we consider the biharmonic equation as follows: Open image in new window where (Delta^{2}u=Delta(Delta u)), (N4), (lambda0), (1 q2) and (muin[0,mu_{0}]), (0mu_{0}infty). The continuous function f verifies the assumptions: ((f_{1})) (f(s)=o( ert s ert )) as (sightarrow0);