Existence Of Nonconstant Positive Solutions For Some Nonlinear Systems With Neumann Boundary Conditions

This dissertation studies the existence of nonconstant positive solutions for nonlinear difference systems and semilinear elliptic systems with Neumann boundary conditions by using fixed point index theory.Meanwhile,we establish the global structure of nonconstant positive radial solutions for the semilinear elliptic systems by applying the bifurcation theory.The main results are described as follows:1.We use the fixed point index theory on cones to study the existence of positive solutions for following nonlinear difference systems with Neumann boundary conditionsFurthermore,the existence of nonconstant positive solutions of the systems was studied by using the fixed point index theory on wedges.T>2 is an integer,f,g:[0,?)x[0,?)?[0,?)are continuous,differentiable and nondecreasing in each variable.In this part,the system considered is the one-dimensional difference form of the system studied by Bonheure et al.in[J.Funct.Anal.,2013].2.Consider the existence of nonconstant positive radial solutions for semilinear elliptic systemswhere(?)is Laplacian operator,BR is the ball of radius R in RN,with N?2.f,g,h:[0,?)×[0,?)×[0,?)?[0,?)are continuous,differentiable and nondecreasing in each variable.Firstly,the fixed point index corresponding to the nondecreasing radial positive solution of the system is obtained through the fixed point index theory on cones.Furthermore,the fixed point index corresponding to the constant solution of the system is obtained through the fixed point index theory on wedges.From the fixed point index of the positive radial solution that is not equal to the fixed point index of the constant solution,we konw that the system has at least one nondecreasing nonconstant positive radial solution.3.We study the global structure of nonconstant nondecreasing positive radial solutions for the semilinear elliptic systemswhere L is Laplacian operator,f,g,h are asymptotically linear growth at infini-ty.Our approach is based on the bifurcation theory due to Dancer.We obtain the branch of the positive solution from a simple eigenvalue through Crandall-Rabinowitz local bifurcation theorem.By virtue of the index jump principle and global bifurcation theory in wedges,thus we determine the direction of the branch of positive solutions and the component is unbounded.Compared with the system studied by Ma et al.in[J.Math.Anal.Appl.,2016],the system considered in this part has more number of equations,so the problem is considered more widely.