Results in Mathematics ( IF 1.1 ) Pub Date : 2021-05-15 , DOI: 10.1007/s00025-021-01426-1 Youpei Zhang
This paper is concerned with the mathematical analysis of solutions to following class of magnetic Kirchhoff problems
$$\begin{aligned} \left\{ \begin{array}{ll} -K\Big (\displaystyle \int _\Omega |\nabla _A u|^2dx\Big )\Delta _Au = g(x,|u|^2)u &{} \quad \mathrm{in} \ \Omega \\ {\,u|_{\partial \Omega }=0}\\ \end{array} \right\} , \end{aligned}$$where \(\Omega \) is a smooth bounded domain in \({{\mathbb {R}}}^N\) (\(N \ge 2\)), A is a magnetic potential, and \(\Delta _A :=(\nabla -iA)^2\) is the magnetic Laplace operator. Additionally, \(K:[0,+\infty )\mapsto (0,+\infty )\) and \(g:{{\bar{\Omega }}}\times {{\mathbb {R}}}\mapsto {{\mathbb {R}}}\) are appropriate continuous functions. Under some natural hypotheses, we obtain the existence of infinitely many solutions in a related magnetic Sobolev space.
中文翻译：
一类具有磁势的基尔霍夫问题的多重性
本文涉及以下一类磁性Kirchhoff问题解的数学分析
$$ \ begin {aligned} \ left \ {\ begin {array} {ll} -K \ Big（\ displaystyle \ int _ \ Omega | \ nabla _A u | ^ 2dx \ Big）\ Delta _Au = g（x， | u | ^ 2）u＆{} \ quad \ mathrm {in} \ \ Omega \\ {\，u | _ {\ partial \ Omega} = 0} \\ \ end {array} \ right \}，\结束{aligned} $$其中\（\ Omega \）是\（{{\ mathbb {R}}} ^ N \）（\（N \ ge 2 \））中的光滑有界域，A是磁势，而\（\ Delta _A：=（\\ nabla -iA）^ 2 \）是磁性拉普拉斯算子。此外，\（K：[0，+ \ infty）\ mapsto（0，+ \ infty）\）和\（g：{{\ bar {\ Omega}}}} \次{{\ mathbb {R}}} \ mapsto {{\ mathbb {R}}} \）是适当的连续函数。在某些自然假设下，我们获得了相关磁Sobolev空间中无限多个解的存在。
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