Research On Solutions For Several Nonlinear Systems Based On Functional Analysis Methods

This paper mainly studies the properties of solutions for six types of nonlinear systems based on fixed point theory and variational methods from functional analysis.To be specific,we define some operators and functionals on some suitable Banach space which are constructed for the desired nonlinear systems.And then,by the fixed point properties of operators or the extreme value properties of functionals,we can get the properties of solutions for the original nonlinear systems.The thesis consists of seven chapters.In chapter one,we introduce some research background and show our new results.In chapter two,we study the existence of solutions for the following integral equa-tion(?)here (?),h is the fractional integral of order ?(0<?<1)with respect to the function h defined by (?)After transferring the above equation to as product operator equation on some suitable function space,we get the existence of solutions for the original equation by applying Darbo fixed point theorem to the product operator.Due to the widely applications of Darbo fixed point theorem,we then generalized this theorem by constructing some suitable contractive functions.In chapter three,we study the existence of solutions for the following integral inclusion system here G is a Caratheodory multi-valued mapping.After define a suitable function space,we transform the above existence problem to a problem of fixed point involved multi-valued mappings.By constructing a class of contractive functions,we generalize the multi-valued Darbo fixed point theorem.And then we apply it to get the existence of solutions for the original integral inclusion system.In chapter four,we consider the existence of solutions for the following nonlinear equation involved 1/2-Laplace operator(?)here the potential function V(x)may change sign,function Q(x)may be unbounded,and nonlinearity f(s)may be discontinuous and satisfies exponential subcritical or critical growth condition.At all,the associated energy functional is not necessary differential,and then we can not use variational methods directly to show the existence of solutions.According to the positive and negative parts of the potential function V(X),we introduce a suitable operator and then transform the original equation to an operator equation.After defining some partial order on suitable function space,we get the existence of solutions for the original equation by applying a fixed point theorem on the Banach semi-lattice.In chapter five,we begin to consider the existence of solutions for the following fractional system by variational methods(?).By some locl redction,we transform the original problem to a local problem.We then show the properties of solutions for the original problem by mountain pass lemma and critical point theory from variational analysis.It is worth mentioning that the original coupled system allows us to consider the general potentials which could not be bounded below by positive constants.We even allow that one of the potentials goes to zero as |x|?? provided the other one goes to infinity at an appropriated rate.Additionally,we should make a delicate analysis on the zero-points intersection set ? of a(x)and b(x)by considering a new boundary value problem,and then get the desired properties in the whole space.In chapter six,we study the existence of multi-bump positive solutions for the following quasi-linear equation in RN here ?N is the N-Laplace defined by (?) is a real parameter.According to the associated auxiliary problem and limit problem,we get the properties of multi-bump solutions on the bottom intV-1(0)by mountain pass lemma and deformation flow.Thus we give a positive answer to the following question:whether the same phenomenon of existence and multiplicity holds,when we consider a quasi-linear equation involved N-Laplace operator and nonlinearity with critical exponential growth inRN.In chapter seven,we consider the existence and concentration of solutions for the following Schrodinger-Kirchhoff equation in(?)in which fRNU=and ?p,A is the p-Laplace operator with magnetic field given by(?)RN?RN is a real vector potential with magnetic field B=? × A,and satisfies Aj?C1(RN,R)for all j=1,2,…,N.By a detailed computation with complex function,we get the properties of solutions on the bottom intV-1(0)by the same methods as in chapter six.At last we point that we do not impose any hypotheses on the behavior of V for |x|?+?.