Lazer-McKenna conjecture for higher order elliptic problem with critical growth

Abstract

This paper is concerned with the following problem involving critical Sobolev exponent and polyharmonic operator: \begin{document}$ \begin{cases} (-\Delta )^mu = u_{+}^{m^*-1}+ \lambda u-s_1 \varphi_1, \hbox{ in } B_1, \\ u \in \mathcal{D}_0^{m,2}(B_1), \end{cases}\;\;\;\;\;\;\;\;\;\;\;\;\;{(P)} $\end{document} where $ B_1 $ is the unit ball in $ \mathbb{R}^{N} $, $ s_1 $ and $ \lambda $ are two positive parameters, $ \varphi_1 > 0 $ is the eigenfunction of $ \left((-\Delta )^m, \mathcal{D}_0^{m,2}(B_1) \right) $ corresponding to the first eigenvalue $ \lambda_1 $ with $ \hbox{ max }_{y \in B_1} \varphi_1(y) = 1 $, $ u_+ = \hbox{ max }(u,0) $ and $ m^* = \frac{2N}{N-2m} $. By using the Lyapunov-Schmits reduction method, we prove that the number of solutions for $ (P) $ is unbounded as the parameter $ s_1 $ tends to infinity, therefore proving the Lazer-McKenna conjecture for the higher order operator equation with critical growth.