$p(x)$-Laplacian方程Robin边界条件下的特征值问题

The Eigenvalue Problem for $p(x)$-Laplacian Equations Involving Robin Boundary Condition

中文关键词:  变指数  特征值  Robin边界条件  $p(x)$-Laplacian方程

英文关键词:variable exponents  eigenvalue  Robin boundary condition  $p(x)$-Laplacian equations

中文摘要:

      本文研究了Robin边界条件下$p(x)$-Laplacian方程特征值问题. 利用变指数Sobolev空间理论, 我们用Luxemburg范数来定义Rayleigh商, 并给出该Rayleigh商的最小值点对应的Euler-Lagrange方程. 根据Ljusternik-Schnirelman原理, 我们证明了Robin边值问题存在无穷多特征值序列, 其中最小的特征值存在且是严格大于零的, 并且与最小的特征值相对应的特征函数不变号.

英文摘要:

      This paper studies the eigenvalue problem for $p(x)$-Laplacian equations involving Robin boundary condition. We obtain the Euler-Lagrange equation for the minimization of the Rayleigh quotient involving Luxemburg norms in the framework of variable exponent Sobolev space. Using the Ljusternik-Schnirelman principle, for the Robin boundary value problem, we prove the existence of infinitely many eigenvalue sequences and also show that, the smallest eigenvalue exists and is strictly positive, and all eigenfunctions associated with the smallest eigenvalue do not change sign.