P-拉普拉斯方程正解和多解的存在性

研究在RN中的一般形式的P-拉普拉斯方程-div(|Du|p-2Du)=f(x,u)(N>P>1).在一定的条件下得到正确和多解.首先建立相应的变分范函,利用H lder和Sobolev不等式证明此范函满足Palais-Smale条件,然后利用爬山原理证明了解的存在性.最后利用严格最大值原理证明了正解的存在性.

Positive and multiple solutions of subcritical P-laplacian equations

The equation -div(|Du|p-2Du)=f(x,u)(N>p>1) on RN as a general equation of the P-laplacian type is considered.Under appropriate conditions,positive and multiple solutions are obtained.We first define a variational functional,and show that the variational functional whose critical points are weak solution of the P-Laplacian equation satisfied the Palais-Smale condition.Then we use the mountain-pass theorem and the strict maximum principle to obtain a nontrivial positive solution.Under the assumption f(x,-u)=-f(x,u), we have the multiple solutions.