On The Existence Of Weak Solutions For A Class Of Fractional Laplace Equations

Partial differential equations are a very important branch in the fields of mathematics,and have very important applications in some subjects, especially in physics. And in partial differential equations, fractional Laplace equations are a class of equations which have very important status, and in the research of fractional Laplace equations, the existence of solutions is an important problem.In the thesis, we mainly discuss the existence of solutions for the following fractional Laplace equation:in the equation, 0 <a <1,(-Δ)~a is the fractional Laplace operator,Ω∈R~N is a bounded domain with Lipschitz boundary, and f: R(?)R, the nonlinearity f is of class with f(0) =0.Firstly, the thesis introduces the research background of the problem, and analyzes the current reaserch situation in domestic and overseas. Next, introducing critical point theory for functionals on partially ordered Hilbert spaces and some other basic definitions and some lemmas, lays the foundation for the main content in our thesis.Lastly, on the foundation of defining the power function we prove that Φ(u) satisfies the(Φ1),(Φ2),(Φ3),(Φ4)(i)(iii) conditions respectly which are provided in the critical point theory, and get the theorem of the existence of weak solutions for the fractional Laplace equation on the top with critical point theory, and give the proof of the theorem.