带Neumann 边界条件的$p(x)$-Kirchhoff 型椭圆系统的研究

本文研究一类带Neumann 边界条件的~$p(x)$-Kirchhoff 型系统解的存在性. 借助于Ekeland变分原理和变指数Sobolev空间理论, 我们给出使得该问题存在解的合适条件.由于Poincar\'{e} 不等式在$W^{1,p(x)}(\Omega)$ 中不再成立, 我们将在$W^{1,p(x)}(\Omega)$ 的某个子空间中证明Poincar\'{e}-Wirtinger 不等式.     

Potential Bifurcation Theorem for a Nonhomogeneous Operator Equation and Its Application北大核心CSCD

The bifurcation phenomenon of the operator equation λ(f1'(x)+f2'(x))=g1'(x)+g2'(x) is studied in this paper. Suppose f2'≡0, f1 and g1 are a-homogeneous, and some other suitable conditions hold, Fučík et al. obtained that each normalized LS-eigenvalue of λf1'(x)=g1'(x) is a bifurcation point of the operator equation above. This paper studies the inhomogeneous case of f1+f2. We establish the same results as theirs when f1, f2, g1 and g2 satisfy some suitable conditions. A Lyusternik-Shnirel'man theorem is obtained as a preliminary result. And for the application of our abstract theorems, the bifurcation phenomenon from arbitrary LS-eigenvalues is studied for a nonlocal elliptic problem. © 2022, Chinese Academy of Sciences. All right reserved.     

带p-Laplacian扰动对称泛函无穷多个临界点的存在性

借助于临界点理论证明了两类扰动拟线性椭圆问题无穷多解的存在性.     

单位球上Yamabe方程的全局分歧

该文研究了N维单位球面S^N上的Yamabe方程■通过分歧的方法,对于任意k≥1,证明了该方程对于任意的λ>λ_k:=(k+N-1)(N-2)/4都至少有一个非常数解v_k,使得v_k-λ⁽1/(N*-1))正好有k个零点,并且它们在(-1,1)中都是单根,其中N^*是Sobolev临界指数.在应用部分,得到了当n≥4时,R^N上非线性椭圆方程非径向解的存在性.此外,还得到了乘积流形中一个流形是单位球时的Yamabe问题的全局分歧结果.     

带跳跃非线性项的p-Laplacian问题的结点解

研究了带跳跃非线性项的p-Laplacian方程结点解的存在性.如果该问题的非线性项跨越其对应齐次问题的Fucik谱,我们证明了该问题至少存在一个结点解.     

一类拟线性特征值问题定号解的精确个数

This paper concerns the exact multiplicity of one-sign solutions of a class of quasilinear elliptic eigenvalue problems with asymptotical nonlinearity at 0 and ∞.The proofs of our main results are based upon bifurcation techniques and stability analysis.