一类超线性椭圆混合边值问题的无穷多解

Existence Results of Infinitely Solutions of Elliptic Mixed Boundary Problems

研究了一类新的椭圆方程混合边值问题,假设非线性项f(x,u)关于u在无穷远处(AR)条件不成立时满足超线性、次临界增长且是奇的,利用对称山路定理证明了该边值问题存在无穷多对弱解.另外还讨论了迹定理和Sobolev嵌入定理在该问题中的应用,几个嵌入不等式被用于定理的证明.

A new elliptic mixed boundary problem is studied. Assume the nonlinear term f（x, u） is superlinear with respect to u at infinity without （AR） condition, sub-critical growth and odd. The existence results of infinitely week solutions are proved by Symmetric Mountain Pass Theorem. Furthermore, trace theorem and Sobolev embedding are discussed and theorem is applied to the mixed boundary problems. Some embedding inequalities are applied to the proof of the theorem.