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On the solvability of singular boundary value problems on the real line in the critical growth case
February  2020, 40(2): 1159-1189. doi: 10.3934/dcds.2020074

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

 Department of Mathematical Science, Tsinghua University, Beijing 100084, China

* Corresponding author: Yuxia Guo

Received  June 2019 Revised  June 2019 Published  November 2019

Fund Project: The first author is supported by NSFC(11771235).

This paper is concerned with the following problem involving critical Sobolev exponent and polyharmonic operator:
 $\begin{cases} (-\Delta )^mu = u_{+}^{m^*-1}+ \lambda u-s_1 \varphi_1, \hbox{ in } B_1, \\ u \in \mathcal{D}_0^{m,2}(B_1), \end{cases}\;\;\;\;\;\;\;\;\;\;\;\;\;{(P)}$
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.
Citation: Yuxia Guo, Ting Liu. Lazer-McKenna conjecture for higher order elliptic problem with critical growth. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 1159-1189. doi: 10.3934/dcds.2020074
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