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

Abstract

<p style='text-indent:20px;'>We consider the existence and multiplicity of nontrivial solutions for a semilinear biharmonic equation with the concave-convex nonlinearities <inline-formula><tex-math id="M1">\begin{document}$ f(x) |u|^{q-1} u $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ h(x) |u|^{p-1} u $\end{document}</tex-math></inline-formula> under certain conditions on <inline-formula><tex-math id="M3">\begin{document}$ f(x), \, h(x) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ p $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ q $\end{document}</tex-math></inline-formula>. Applying the Nehari manifold method along with the fibering maps and the minimization method, we study the effect of <inline-formula><tex-math id="M6">\begin{document}$ f(x) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ h(x) $\end{document}</tex-math></inline-formula> on the existence and multiplicity of nontrivial solutions for the semilinear biharmonic equation. When <inline-formula><tex-math id="M8">\begin{document}$ h(x)^+ \neq 0 $\end{document}</tex-math></inline-formula>, we prove that the equation has at least one nontrivial solution if <inline-formula><tex-math id="M9">\begin{document}$ f(x)^+ = 0 $\end{document}</tex-math></inline-formula> and that the equation has at least two nontrivial solutions if <inline-formula><tex-math id="M10">\begin{document}$ \int_\Omega |f^+|^r\, \text{d}x \in (0, \varLambda^r) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M11">\begin{document}$ r $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M12">\begin{document}$ \varLambda $\end{document}</tex-math></inline-formula> are explicit numbers. These results are novel, which improve and extend the existing results in the literature.</p>