We consider the existence and multiplicity of nontrivial solutions for a semilinear biharmonic equation with the concave-convex nonlinearities $ f(x) |u|^{q-1} u $ and $ h(x) |u|^{p-1} u $ under certain conditions on $ f(x), \, h(x) $, $ p $ and $ q $. Applying the Nehari manifold method along with the fibering maps and the minimization method, we study the effect of $ f(x) $ and $ h(x) $ on the existence and multiplicity of nontrivial solutions for the semilinear biharmonic equation. When $ h(x)^+ \neq 0 $, we prove that the equation has at least one nontrivial solution if $ f(x)^+ = 0 $ and that the equation has at least two nontrivial solutions if $ \int_\Omega |f^+|^r\, \text{d}x \in (0, \varLambda^r) $, where $ r $ and $ \varLambda $ are explicit numbers. These results are novel, which improve and extend the existing results in the literature.

中文翻译：

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具有权函数的半线性双调和方程非平凡解的存在性和多重性

我们考虑具有凹凸非线性的半线性双调和方程的非平凡解的存在性和多重性 $ f(x) |u|^{q-1} u $ 和 $ h(x) |u|^{p-1 } u $ 在 $ f(x), \, h(x) $, $ p $ 和 $ q $ 上的某些条件下。应用 Nehari 流形方法以及纤维映射和最小化方法，我们研究了 $ f(x) $ 和 $ h(x) $ 对半线性双调和方程非平凡解的存在性和多重性的影响。当 $ h(x)^+ \neq 0 $ 时，如果 $ f(x)^+ = 0 $，我们证明方程至少有一个非平凡解，如果 $\int_\，方程至少有两个非平凡解Omega |f^+|^r\, \text{d}x \in (0, \varLambda^r) $，其中 $ r $ 和 $ \varLambda $ 是显式数字。这些结果是新颖的，它们改进和扩展了文献中的现有结果。