
Minimax Theory and its Applications 08 (2023), No. 2, 381392 Copyright Heldermann Verlag 2023 Multiplicity Theorems for Biharmonic KirchhoffType Elliptic Problems Lingju Kong Dept. of Mathematics, University of Tennessee, Chattanooga, U.S.A. lingjukong@utc.edu [Abstractpdf] We study the existence of multiple weak solutions for the biharmonic Kirchhofftype elliptic problem \begin{equation*} \left\{\ \begin{aligned} & M\left(\int_{\Omega}(\Delta u^p+\nabla u^p)dx\right)\left(\Delta_p^2u\Delta_p u\right) =\sum_{i=1}^k\alpha_i(x)f_i(u) + \gamma(x)\ \text{in}\ \Omega,\\[2pt] & u = \Delta u=0\quad \text{on}\ \partial\Omega,\ \ \int_{\Omega}\left(\Delta u^{p}+\nabla u^p\right)dx<\rho. \end{aligned} \right. \end{equation*} We establish necessary and sufficient conditions on $f_i$, $i=1,\ldots, k$, under which there exists functions $\alpha_i, \gamma\in C(\overline{\Omega})$, $i=1,\ldots, k$, such that the above problem has at least two weak solutions. Our proof uses the variational approaches and relies on an existence result for crical points of functionals in Banach spaces recently obtained by Ricceri. Keywords: Kirchhofftype problems, $p$Laplacian operator, pbiharmonic operator, weak solutions, critical points, contraction mapping theorem. MSC: 35G30, 35J58, 49J35. [ Fulltextpdf (128 KB)] for subscribers only. 