Three Solutions for the Parabolic Problem with Jumping Nonlinearity Crossing One Eigenvalue

초록

Let be a bounded open subset of Rn with smooth boundary. In this paper we consider the multiplicity of the solutions of the following parabolic boundary value problem Dtu = Lu + bu+ &#8722; au&#8722; &#8722; s1 &#8722; h(x) in × R, u(x, t) = 0, x 2 @ , t 2 R. u(x, t) = u(x, t + 2) in × R. Here L be the self-adjoint strongly elliptic partial differential operator. Let k be the infinitely many eigenvalues and k, k = 1, 2, . . . be the associated normalized eigenfunctions with 0 < 1 < 2 · · · k ! 1 and 1 > 0 of the eigenvalue problem Lu + u = 0 in , u = 0 on @. The physical model for this kind of the jumping nonlinearity problem can be furnished by traveling waves in suspension bridges.