一类含临界增长项的 Kirchhoff 型四阶椭圆方程解的存在性——献给李工宝教授 70 寿辰

Abstract

Abstract:

In this paper, we consider the existence of solutions to a class of fourth-order Kirchhoff-type elliptic equations with critical term and linear pertubation

$\begin{aligned} &\Delta^2 u-\bigg(a+b\int_{\mathbb{R}^N}|\nabla u|^2{\rm d}x\bigg)\Delta u=\lambda|u|^{2^\#-2}u+\sigma h(x),\;x\in\mathbb{R}^N,\\ &u\in\mathcal{D}^{2,2}(\mathbb{R}^N), \end{aligned}$

where $\displaystyle2^\#=\frac{2N}{N-4}$ is the critical Sobolev exponent. With the help of the Concentration Compactness Principle, Ekeland's Variational Principle and Mountain Pass Lemma, we show that the (P.S.)$_c$ condition is locally satisfied and then obtain at least two nontrivial weak solutions under some assumptions on $a,\lambda$ and $\sigma$.

Key words: Kirchhoff-type Equations, Concentration Compactness Principle, Mountain Pass Lemma

CLC Number:

O175.59

Xiaochun Liu, Liwei Wang. The Existence of Solutions to a Class of Fourth-Ordered Kirchhoff-Type Equations with Critical Growth[J].Acta mathematica scientia,Series A, 2025, 45(6): 1752-1767.

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The Existence of Solutions to a Class of Fourth-Ordered Kirchhoff-Type Equations with Critical Growth

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