应用Ricceri的临界点定理证明p-Laplacian方程的三解问题

边值问题的提出和发展,与流体力学、材料力学、波动力学以及核物理学等密切相关,并且在现代控制理论等学科中有重要应用.其中边值问题的形式多种多样,并且对于边值问题的存在性的证明也包括很多种方法.首先,介绍了基本临界点问题的背景.然后,阐述了Ricceri的临界点定理及其推论.其次,研究一类带有p-Laplace算子的2点边值问题,应用Ricceri的临界点定理证明了这个边值问题解的存在性,并且把Ricceri的临界点定理从证明对称的边值条件扩展到可证明非对称的边值条件,化简了所需要的限制条件.最后用实例验证了所得结果的可行性.

Three solutions of p-Laplacian equations via critical point theorem of Ricceri

The development of the boundary value problem is related to fluid mechanics, material mechanics, wave mechanics and nuclear physics, and it is important for a modern control theory. There are many forms for boundary value problems, and it also includes a lot of methods for the proof of the existence of boundary value problems. The first part describes the critical point theorem in the background. The second part describes Ricceri critical point theorems and its inferences. The third part studies a class of two-point boundary value problems with p-Laplace operator, and it applies the critical point theorem of Ricceri to prove that the boundary value problems. This paper make the symmetric boundary conditions extend asymmetric boundary conditions, and then simplify the restrictions. An example is given to demonstrate our main result.