Research On The Existence Of Solutions For Some Quasilinear Schro(?)dinger Equations

In this thesis,by using the method of change of variable and variational meth-ods,we study the existence of nontrivial solutions for several quasilinear Schrodinger equations in the whole space RN.Specifically,we first consider the following equationin Chapters 2 and 3,where N?3,V?C(RN,g+)is bounded potential,g?C1(R,R)satisfying some conditions.We prove the existence result of positive solutions of the equation(G)via change of variable and the mountain pass theorem for the nonlinearity h?C(R,R)either is superlinear about t at infinity or asymptotically linear about t at infinity.Moreover,by using the above result,together with a cut-off technique and the Morse iteration technique,we obtain two existence results for two peculiar equation from the equation(G),which supplement and extend the corresponding results in[4,61].Next,in Chapter 4 we consider the following equationwith the potential V(x)being radial symmetry and the nonlinearity h(t)being super-linear about t at infinity.By constructing a special change of variable and applying a perturbed critical point theorem,we prove that the preceding equation has a nontrivial solution,which extends the corresponding result in[61].In Chapter 5,by a change of variable,together with a splitting lemma and the linking theorem,we get the existence of positive solution of the following equationfor ?>0,the nonlinearity h(t)is asymptotically 3-linear about t at infinity and V?C(RN,R+)is a potential satisfying some assumptions.In particular,we also get the existence of positive solutions of the following equationOur results improve the corresponding results in[19,80].Finally,in Chapter 6,via a change of variable and the mountain pass theorem,we show that for each ?>0 there exists a ?1(?)>0 such that the following equationhas a positive solution for all ?>?1(?),where both the potentials V(x),K(x)are vanishing at infinity,nonlinearity term h?C(R,R),which extends corresponding results in[4].