Existence and multiplicity of nontrivial solutions for a semilinear biharmonic equation with weight functions

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