We consider the boundary value problem

$$\begin{aligned} \begin{array}{rcl} -\Delta _p u &{} = &{} \alpha |u|^{p-2}u^+-\beta |u|^{p-2}u^- \text{ in } \Omega ,\\ u &{} = &{} 0 \text{ on } \partial \Omega , \end{array} \end{aligned}$$

where \(\Delta _p u:=\nabla \cdot (|\nabla u|^{p-2}\nabla u)\) for \(1<p<\infty \), \(\Omega \) is a smooth bounded domain in \(\mathbb {R}^N,u^{\pm }:=\max \{\pm u,0\}\), and \((\alpha ,\beta )\in \mathbb {R}^2\). If this problem has a nontrivial weak solution, then \((\alpha ,\beta )\) is an element of the Fučík Spectrum, \(\Sigma \subset \mathbb {R}^2\). Our main result is to provide a variational characterization for a class of curves in \(\Sigma \).

中文翻译：

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p -Laplace算子的Fučík谱的扩展变分刻画

我们考虑边值问题

$$ \ begin {aligned} \ begin {array} {rcl}-\ Delta _p u＆{} =＆{} \ alpha | u | ^ {p-2} u ^ +-\ beta | u | ^ {p -2} u ^-\ text {in} \ Omega，\\ u＆{} =＆{} 0 \ text {on} \ partial \ Omega，\ end {array} \ end {aligned} $$

其中\（\德尔塔_p U：= \ nabla \ CDOT（| \ nabla U | ^ {P-2} \ nabla U）\）为\（1 <P <\ infty \） ，\（\欧米茄\）是\（\ mathbb {R} ^ N，u ^ {\ pm}：= \ max \ {\ pm u，0 \} \）中的光滑有界域，以及\（（\ alpha，\ beta）\ mathbb {R} ^ 2 \）。如果此问题具有非平凡的弱解，则\（（\ alpha，\ beta）\）是Fučík频谱的元素\（\ Sigma \ subset \ mathbb {R} ^ 2 \）。我们的主要结果是为\（\ Sigma \）中的一类曲线提供变化特征。