PARABOLIC PROBLEM WITH ASYMPTOTICAL LINEARITY

초록

Let ­ be a bounded open subset of Rn with smooth boundary @­. We consider the multiplicity of the solutions of the following parabolic boundary value problem Dtu = Lu + bu+ ¡ au¡ ¡ s&Aacute;1 ¡ h(x) in ­ &pound; R; (1:2) u(x; t) = 0; x 2 @­; t 2 R; u(x; t) = u(x; t + 2¼); in ­ &pound; R: Here L is the self-adjoint strongly elliptic partial differential operator. The eigenvalue problem Lu+¸u = 0 in ­ with Dirichlet boundary condition has infinitely many eigenvalues ¸k and the associated normalized eigenfunctions &Aacute;k, k = 1; 2; : : : ; with 0 < ¸1 < ¸2 · &cent; &cent; &cent; · ¸k ! 1 and &Aacute;1 > 0. Theorem 1.1. Assume that ¡1 < a < ¸1 < b < ¸2 and s > 0. Then (1.1) has at least three periodic solutions.