$ \mathbb{R}^4 $ 中一类带陡峭位势的临界 Kirchhoff 型方程的基态解

Abstract

Abstract:

In this paper, we focus on dealing with a class of critical Kirchhoff type equation

$\left\{\begin{array}{ll}\displaystyle-\left(a+b\int_{\mathbb{R}^4} |\nabla u|^2\mathrm{d}x\right)\Delta u+\lambda V(x)u =|u|^{2}u +f(u) &\text{ in } \mathbb{R}^4,\\u\in H^{1} (\mathbb{R}^4),\end{array}\right.$

where $ a,b > 0$ are constants and $ \lambda > 0 $. The nonlinear growth of $ |u|^{2}u $ reaches the Sobolev critical exponent since $ 2^{*}= 4 $ in dimension 4. Assume that $ V $ is the nonnegative continuous potential, which represents a potential well with the bottom $ V^{-1}(0) $ and $ f \in C(\mathbb{R},\mathbb{R}) $ satisfies suitable conditions. By the variational methods, the existence of at least a ground state solution is obtained. Moreover, we study the concentration behavior of the ground state solutions as $ \lambda \rightarrow\infty $ and their asymptotic behavior as $ b\rightarrow 0 $ and $ \lambda \rightarrow\infty $, respectively.

Key words: Kirchhoff type equation, critical growth, variational methods, steep potential well, ground state solutions

CLC Number:

O177.91

Chen Zhengyan,Zhang Jiafeng. Ground State Solutions for a Class of Critical Kirchhoff Type Equation in $ \mathbb{R}^4$ with Steep Potential Well[J].Acta mathematica scientia,Series A, 2025, 45(2): 450-464.

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References 24

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Ground State Solutions for a Class of Critical Kirchhoff Type Equation in $ \mathbb{R}^4$ with Steep Potential Well

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