Degree Theory Applied to the Variable Coefficient Semilinear Biharmonic Problem

초록

In this paper we consider the multiplicity result for the following biharmonic equation with the variable coefficient semilinear term and Dirichlet boundary condition. Here is the Laplace operator and 1 is the positive eigenfunction corresponding to the first eigenvalue of the eigenvalue problem u + cu &#8722; b(x)u = μu with Dirichlet boundary condition. Our main result is as follows: Theorem 1. Let c < 1 and n(n &#8722; c) < b(x) < n+1(n+1 &#8722; c), n 1. Then there exists s0 < 0 such that for any s with 0 < s s0 if n is even then (1) has at least three solutions, one of which is a positive solution, and if n is odd then (1) has at least two solutions, one of which is a positive solution.