$p(x)$-Laplacian方程Robin边界条件下的特征值问题

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.

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

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.