We prove the existence of a positive solution for the problem \({\rm{\gamma}}{{\rm{\Delta}}^2}u - m\left(u \right){\rm{\Delta}}u = \mu a\left(x \right){u^q} + b\left(x \right){u^p},\,\,{\rm{in}}\,{\rm{\Omega ,}}\,\,\,\,\,u = {\rm{\gamma \Delta}}u = 0,\,\,{\rm{on}}\,\,\partial {\rm{\Omega ,}}\) where Ω ⊂ ℝ^N is a bounded smooth domain, γ ∈ {0, 1},0 < q > 1 < p, m is weakly continuous in \({H^2}\left({\rm{\Omega}} \right) \cap H_0^1\left({\rm{\Omega}} \right),a \in {L^\infty}\left({\rm{\Omega}} \right)\) is nonnegative and b is a bounded potential which can change sign. The solution is obtained via a sub-supersolution approach when the parameter µ > 0 is small.

我们证明存在针对问题\（{\ rm {\ gamma}} {{\ rm {\ Delta}} ^ 2} u-m \ left（u \ right）{\ rm {\ Delta} } u = \ mu a \ left（x \ right）{u ^ q} + b \ left（x \ right）{u ^ p}，\，\，{\ rm {in}} \，{\ rm { \ Omega，}} \，\，\，\，\，u = {\ rm {\ gamma \ Delta}} u = 0，\，\，{\ rm {on}} \，\，\ partial {\ RM {\欧米茄，}} \） ，其中Ω⊂ℝ ^ñ是有界光滑区域，γ＆Element; {0，1}，0 < q > 1 < p，M是在弱连续\（{H ^ 2} \左（{\ rm {\ Omega}} \ right）\ cap H_0 ^ 1 \ left（{\ rm {\ Omega}} \ right），\ in {L ^ \ infty} \ left（{\ rm {\ Omega }} \ right）\）是非负的，b是可以改变符号的有限势。当参数μ时，可通过子超解法获得解。 > 0很小。