Multiple sign-changing solutions to the Sturm-Liouville boundary value problem with resonance

作者：Zhang Qi; Li Fuyi; Zhu Xiaoli

来源：Applied Mathematics and Computation, 2012, 219(3): 1061-1072.

DOI：10.1016/j.amc.2012.07.012

• Sturm-Liouville boundary value problem
• Sign-changing solution
• Topological degree
• Fixed point index
• Critical group

In this paper, we study the Sturm-Liouville boundary value problem -(p(x)u';)'; + q(x)u = f(x; u), x is an element of [0; 1] subject to alpha u(0) - beta u';(0) = gamma u(1) + delta u';(1) = 0. By constructing a new Sobolev space H-alpha,gamma(1) [0; 1], we discuss the existence of multiple solutions, especially the existence of multiple sign-changing solutions to this problem when the nonlinear f is resonant both at 0 and infinity. By combining the methods of the Morse theory, the topological degree and the fixed point index, we establish a multiple solutions theorem which guarantees that the problem has at least six nontrivial solutions. If this problem has only finitely many solutions then, of these solutions, there are two positive solutions, two negative solutions and two sign-changing solutions.

• 出版日期2012-10-15
• 单位山西大学