On Signed And Sign-changing Solutions For Several Classes Of Nonlinear Problems

In this paper, we mainly use the variational methods and some critical point theorems to study the existence and multiplicity of mountain pass type signed and sign-changing solutions for several classes of nonlinear problems. This article is divided into four chapters, the first chapter introduces some background knowledges and research contents of this paper, the main contents of the remaining chapters are as follows:The second chapter studies the following Kirchhoff-type equation The Kirchhoff-type equation has been closely attented, has wide application in many neighborhood. As far as we know, there have been many papers concerned with the existence of sign-changing solutions of the Kirchhoff-type equations in bounded domain. We studies the existence of mountain pass type positive solution, negative solution and infinitely many sign-changing solutions in R3.The third chapter studies the following nonlinear Schrodinger equation we obtain the existence and multiplicity of mountain pass type signed and sign-changing solutions. In [22], when the nonlinear term h satisfies the appropriate growth conditions and the global (AR) condition, the author proved the existence of sign-changing solution; In [105], the author proved that the nonlinear Schrodinger equation had a positive solution, a negative solution and infinitely many sign-changing solutions. Different from the literature [22] and [105], under the local (AR) condition and the weak conditions we prove the existence of mountain pass type positive solution, negative solution and infinitely many sign-changing solutions to the nonlinear Schrodinger equation. The results of this chapter improve the results in [22] and [105].The fourth chapter study respectively the quasilinear elliptic equation the following quasilinear Schrodinger equation and the following generalized quasilinear Schrodinger equation we obtain the existence of the mountain pass type positive solution, negative solution, sign-changing solution and infinitely many sign-changing solutions. Due to the existence of the quasilinear term, the corresponding energy functionals of these problems are not well defined in the familiar working space, thus we can not apply directly the variational methods to study these problems. To overcome this difficulty, we make the change of variable, these quasilinear problems are transformed to semilinear problems and the familiar Sobolev space is corresponding functionals’ workspace. Under the proper conditions, we obtain the existence of the mountain pass type positive solution, negative solution and sign-changing solution. Furthermore, if the nonlinear term is odd with respect to its second variable, then these quasilinear elliptic equations possess infinitely many sign-changing solutions.