This paper is concerned with the following elliptic problem:

$$\left\{\begin{array}{c}{(-\mathrm{\Delta })}^{m}u=u_{+}^{m^{\ast }-1}+\lambda u-s_1{\phi }_1,\text{in}{B}_1,\hfill \\ u\in {\mathcal{D}}_0^{m,2}\left(B_1\right),\hfill \end{array}\right.(\mathrm{P})$$(P)

where ${(-\mathrm{\Delta })}^m$ is the polyharmonic operator, $m^{\ast }=\frac{2N}{N-2m}$ is the critical Sobolev embedding exponent. B₁ is the unit ball in ${\mathbb{R}}^N$, s₁ and $\lambda >0$ are parameters, ${\phi }_1>0$ is the eigenfunction of $\left({(-\mathrm{\Delta })}^m,{\mathcal{D}}_0^{m,2}\left(B_1\right)\right)$ corresponding to the first eigenvalue λ₁ with ${\max }_{y\in B_1}{\phi }_1(y)=1$, $u_{+}=\max (u,0)$. By using the Lyapunov–Schimit reduction method combining with the minimax argument, we construct the solutions to (P) with many peaks near the boundary but not on the boundary of the domain. Moreover, we prove that the number of solutions for the problem (P) is unbounded as the parameter tends to infinity, therefore proving the Lazer–McKenna conjecture for the higher order case with critical growth.

中文翻译：

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具有临界增长的多谐方程的多峰解

本文涉及以下椭圆问题：

$$\left\{\begin{array}{c}{（-\mathrm{\Delta }）}^{米}ü={ü}_{+}^{{米}^{\ast }-\mathrm{1个}}+\lambda ü-s_{\mathrm{1个}}{\phi }_{\mathrm{1个}}，\text{在}{乙}_{\mathrm{1个}}，\hfill \\ ü\in {\mathcal{d}}_0^{米，2}（{乙}_{\mathrm{1个}}），\hfill \end{array}\right.（\mathrm{P}）$$（P）

在哪里 ${（-\mathrm{\Delta }）}^{米}$ 是多谐波运算符， ${米}^{\ast }=\frac{2ñ}{ñ-2米}$是Sobolev嵌入的关键指数。B ₁是单位球${\mathbb{[R}}^{ñ}$，s ₁和$\lambda >0$ 是参数 ${\phi }_{\mathrm{1个}}>0$ 是...的本征函数 $\left({（-\mathrm{\Delta }）}^{米}，{\mathcal{d}}_0^{米，2}（{乙}_{\mathrm{1个}}）\right)$对应于所述第一特征值λ ₁与${\mathrm{最大限度}}_{ÿ\in {乙}_{\mathrm{1个}}}{\phi }_{\mathrm{1个}}（ÿ）=\mathrm{1个}$， ${ü}_{+}=\mathrm{最大限度}（ü，0）$。通过使用Lyapunov–Schimit约简方法与minimax参数相结合，我们构造了（P）的解决方案，该解决方案在边界附近但不在边界处具有许多峰。此外，我们证明了随着参数趋于无穷大，问题（P）的解数是无穷大的，因此证明了Lazer–McKenna猜想对于具有临界增长的高阶情况。