On An Eigenvalue Problem For An Anisotropic Elliptic Equation

We study the existence of infinitely many solutions for anisotropic variable exponent problem of the type {-Sigma(N)(i=1) partial derivative xiai(x,partial derivative xiu) + Sigma i-1Nai(x,u) = lambda vertical bar u vertical bar q(x)-2u in Omega, partial derivative u/partial derivative v = 0 on partial derivative Omega (1) Where Omega subset of R-N(N >= 3) is a bounded domain with smooth boundary partial derivative Omega, lambda > 0,, p(i),q are continuous functions on (Omega) over bar such that p(i)(x) >= 2, for all(x) is an element of Omega and i is an element of {1, 2, ....., N}. The main result of this paper establishes the existence of two positive constants lambda(0) and lambda(1) with lambda(0) <= lambda(1) such that every lambda is an element of(lambda(1), infinity) is an eigenvalue, while no lambda is an element of(0, lambda(0)) can be an eigenvalue of the above problem.