分数阶Laplace方程组的山路解

对一类非线性分数阶Laplace方程组Dirichlet问题非平凡解以及正解的存在性分别进行了研究.针对非线性分数阶Laplace方程组在满足Dirichlet边值条件下所具有的特征,通过定义能量空间,然后在该空间中利用Sobolev嵌入定理、控制收敛定理、Brezis-Leb引理,证明分数阶方程组的能量泛函满足Palais-Smale紧性条件,最后利用分数阶Sobolev空间中的山路引理,得出方程组存在非平凡临界点,也即得出这类非线性分数阶Laplace方程组Dirichlet问题存在非平凡解的结论.此外,还利用Nehari流形、极小能量法,通过比较能量法得出一类耦合的非线性分数阶Laplace方程组Dirichlet问题存在正解需要满足的条件,进而得出这类分数阶Laplace方程组存在正解的结论.

Mountain Pass Solutions for the System of Equations Driven by the Fractional Laplace Operator

The existence of nontrivial solutions and positive solutions for fractional Laplace systems with Dirichlet boundary conditions was investigated.According to the features when the Dirichlet boundary conditions of fractional Laplace systems are satisfied and based on the mountain pass theorem, embedding theorem, dominated convergence theorem and Brezis-Leb lemma in Sobolev space, the Palais-Smale compactness conditions of the energy functional were proved.Then the existence of nontrivial solutions for fractional Laplace systems with Dirichlet boundary conditions was concluded.Moreover, The Nehari manifold and minimal energy method were used to prove the existence of positive solutions for fractional Laplace systems with Dirichlet boundary conditions.