Research On The Existence Of Solutions For Some Nonlinear Differential Equations

Nonlinear analysis is based on all sorts of nonlinear problems in real world, it is the cornerstone of all kinds of nonlinear differential equation, and it consists of partial order theory, topological degree theory, critical point theory and so on. In our paper, we investigate the existence and multiplicity of solutions for several classes of nonlinear differential equations by using topological degree theory and critical point theory.The dissertation is divided into five chapters. In Chapter 1,we introduce some basic concepts and theorems in nonlinear analysis. In Chapter 2, we first study the existence of weak solutions for the following fractional Schrodinger equation where N≥2, s∈(0,1),(-Δ)s stands for the fractional Laplacian,f∈C(RN× R,R).Secondly, we also investigate the fractional Schrodinger equation: where N, s,(-Δ)s are as above, g(x) is a perturbation function, In view of this, existence of at least one weak solution is discussed. Also we consider the effect of the parameter λ and the perturbation term g on the existence of solutions.In Chapter 3, we investigate the stationary solutions for the generalized Kadomtsev-Petviashvili equation in bounded domain in Rn where ds denotes the inverse operator, the inner prod-uct We mainly utilize the mountain pass theorems and the fountain theorems to obtain the existence of infinitely many weak solutions for the above problem.In Chapter 4, we are devoted to the existence of weak solutions for the fourth-order Navier boundary value problem where λ is a parameter, Δ2 denotes the biharmonic operator,Ω(?)RN(N>4) is a smooth bounded domain,f∈C(R,R). Moreover, under the famous Landesman-Lazer condition, by using of (S)+ degree theory, the existence of weak solutions for the problem is also obtained, where f∈C(R,R),g∈Lq(Ω),q∈(2N/N+4,+∞].In Chapter 5,we investigate positive solutions for the boundary value prob-lem of fractional order involving Riemann-Liouville’s derivative where α∈ (2,3], f:[0,1]×[0,+∞)×[0,+∞)×[0,+∞)â†'(-∞,+∞)is contin-uous, and f(t, x1, x2, x3) is bounded below, i.e., there exists a real number M>0 such that f(t, x1, x2,x3)+M≥ 0,(?)t ∈[0,1],xi∈[0,+00), i=1,2,3. The inter-esting point lies in the fact that the nonlinear term is allowed to depend on u, u’ and-D0α+u.