带变号权的$p$ -Laplacian方程的单侧全局分支和节点解
Unilateral global bifurcation and nodal solutions for the $p$-Laplacian with sign-changing weight
摘要：
本文建立了一类变号权拟线性椭圆型方程的Dancer型单边全局分支结果。在扰动函数的某些自然假设下，我们证明了$(μ_k^ν(p),0)$是上述问题的一个分歧点，并且存在两个不同的无界连续统$(\mathcal{C}_{k}^ν)^+$和$(\mathcal{C}_{k}^ν)^-$，它们由$(μ_k^ν(p), 0)$的分歧分支$\mathcal{C}_{k}^ν$组成，其中$μ_k^ν(p)$是与上述问题对应的线性问题$ν\in\{+,-\}$的第$k$个正或负特征值.作为上述单侧全局分歧结果的应用，我们研究了一类变号权拟线性椭圆型方程节点解的存在性。此外，基于Drábek和Huang（1997）[ \ref{DH} ]的分歧结果，我们研究了一类变号权的高维拟线性椭圆型问题的单符号解的存在性.
英文摘要：
In this paper, we shall establish a Dancer-type unilateral global bifurcation result for a class of quasilinear elliptic problems with sign-changing weight. Under some natural hypotheses on perturbation function, we show that $(μ_k^ν(p),0)$ is a bifurcation point of the above problems and there are two distinct unbounded continua, $(\mathcal{C}_{k}^ν)^+$ and $(\mathcal{C}_{k}^ν)^-$, consisting of the bifurcation branch $\mathcal{C}_{k}^ν$ from $(μ_k^ν(p), 0)$, where $μ_k^ν(p)$ is the $k$-th positive or negative eigenvalue of the linear problem corresponding to the above problems, $ν\in\{+,-\}$. As the applications of the above unilateral global bifurcation result, we study the existence of nodal solutions for a class of quasilinear elliptic problems with sign-changing weight. Moreover, based on the bifurcation result of Drábek and Huang (1997) [\ref{DH}], we study the existence of one-sign solutions for a class of high dimensional quasilinear elliptic problems with sign-changing weight.
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