Existence Of Solutions For Some Kinds Of Nonlinear Fractional Laplace Equations

In the past decades,the classical Laplace equation theory has been fully developed.This kind of partial differential equations has important applications in many fields,such as mathematics,physics,chemistry,biology,engineering,materials and so on.The Laplace equations can well explain the diffusion phenomena in many practical problems,but it can not explain many anomalous diffusion phenomena.Therefore,fractional Laplace equations have been developed.A fractional Laplace operator is a generalization of a Laplace operator and inherits some important properties of the Laplace operator,such as bounded linearity and selfadjoint property,which provides convenience for the research of fractional Laplace equations.However,unlike a Laplace operator,a fractional Laplace operator is a nonlocal quasi-differential operator,which has caused substantial difficulties in the research of related problems.In this paper,taking the fractional Sobolev spaces as the framework and using the properties of fractional Laplace operators,we study the existence,multiplicity and bifurcation of solutions for several classes of nonlinear fractional Laplace equations.Firstly,we study a class of fractional Laplace equations with indefinite nonlinear terms.By applying a basic inequality,we obtain the corresponding weak maximum principle.Using the weak maximum principle,we get the upper and lower solutions method of the equations.By some complicated analysis techniques,we establish the Hopf Lemma corresponding to the equations.Based on the spectral theory of fractional Laplace operators,we obtain what the parameters should satisfy.Applying the upper and lower solutions method of the equations,we prove that all the parameters which make the equations have solutions constitute an interval.Through the constraint extremum method,we obtain the left endpoint of this interval is the first eigenvalue of the fractional Laplace operator.Secondly,we study the existence of positive solutions for a class of nonhomogeneous semilinear fractional Laplace equations.Through some analysis techniques,we obtain the corresponding weak and strong maximum principles.Similar to the Laplace equations,we give a prior estimate and regularity analysis of the nonnegative solutions of the equations.We generalize the concentration-compactness principle of Lions to these equations.Then we reduce the existence of solutions of the equations to the existence of critical points of the corresponding functionals.Using the Ekeland’s variational principle,we obtain the existence of solutions of the equations.By applying the iterative method of upper and lower solutions,we get the existence of minimal positive solutions and the monotonicity of minimal positive solutions.Finally,we continue to discuss the multiplicity and bifurcation of positive solutions of the nonhomogeneous semilinear fractional Laplace equations.Based on the minimal positive solutions,we reduce the existence of second positive solutions of the equations to the existence of critical points of the corresponding functionals.By adding certain conditions,we get Aubin type estimates for the functionals.We establish the concentrationcompactness principle of these equations,and then use the Mountain Pass Lemma to obtain the second positive solutions of these equations.By adding the convexity,we get the uniqueness of positive solution of an equation at the critical value of the parameter.Using the critical value and the unique positive solution as the original point pair and applying the Bifurcation Theorem of Crandall and Rabinowitz,we obtain the bifurcation property of the equation in the neighborhood of the original point pair.We also give some properties of the set of all the positive solutions of these equations.The results obtained in this paper are the generalization of Laplace equation.However,from the proofs of these results,we can see that the discussion of fractional Laplace equations is obviously different from that of Laplace equations.