In this paper, we consider the following modified quasilinear problem: − Δ u− κ uΔ u2=λ a(x)u− α +b(x)uβ inΩ ,u> 0inΩ ,u=0on∂ Ω , $$\begin{array}{} \left\{\begin{array}{c}\, -{\it\Delta} u-\kappa u{\it\Delta} u^2 = \lambda a(x)u^{-\alpha}+b(x)u^\beta \, \, in\, {\it\Omega}, \\\!\! u \gt 0 \, \, in\, {\it\Omega}, \, \, \, \, \, \, \, u = 0 \, \, on \, \partial{\it\Omega} , \\ \end{array}\right. \end{array} $$ where Ω ⊂ ℝ N is a smooth bounded domain, N ≥ 3, a , b are two bounded continuous functions, α > 0, 1 < β ≤ 22 * − 1 and λ > 0 is a bifurcation parameter. We use the framework of analytic bifurcation theory to obtain an analytic global unbounded path of solutions to the problem. Moreover, we get the direction of solution curve at the asmptotic point.

中文翻译：

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修正的负指数拟线性方程的分岔分析

在本文中，我们考虑以下修正的拟线性问题： − Δ u− κ uΔ u2=λ a(x)u− α +b(x)uβ inΩ ,u> 0inΩ ,u=0on∂ Ω , $$\begin {数组}{} \left\{\begin{数组}{c}\, -{\it\Delta} u-\kappa u{\it\Delta} u^2 = \lambda a(x)u^{ -\alpha}+b(x)u^\beta \, \, in\, {\it\Omega}, \\\!\! u \gt 0 \, \, in\, {\it\Omega}, \, \, \, \, \, \, \, u = 0 \, \, on \, \partial{\it\Omega} , \\ \end{数组}\对。\end{array} $$ 其中 Ω ⊂ ℝ N 是光滑有界域，N ≥ 3, a , b 是两个有界连续函数，α > 0, 1 < β ≤ 22 * - 1 和 λ > 0 是分岔范围。我们使用解析分岔理论的框架来获得解决问题的解析全局无界路径。此外，我们得到了渐近点处解曲线的方向。