Quasilinear schrodinger equation pdf

The timeindependent schroedinger equation a very important special case of the schroedinger equation is the situation when the potential energy term does not depend on time. Liu was visiting the university of notre dame, he wants to thank the mathematical department of the university for the hospitality, and the china scholarship council for financial support. In fact, this particular case will cover most of the problems that well encounter in ee 439. The description of nature is essentially probabilistic, with the probability of an. Introduction and statement of main results this article concerns the singular quasilinear schrodinger equation with the dirichlet boundary value condition. Such equations have been derived as models of several physical phenomena. Soliton solutions for quasilinear schrodinger equations, i. Existence and concentration of positive solutions for. Quasilinear schrodinger equation, ljusternikschnirelmann theory, positive solutions, critical problems. A system is completely described by a wave function. Schrodinger equation with a cubic nonlinearity, schrodinger equation with a powerlaw nonlinearity. Multiple positive solutions for a quasilinear system of schrodinger. This project is supported by national natural science foundation of china grant no.

We achieved our results by using minimax methods and lusternikschnirelman theory of critical points. Soliton solutions for quasilinear schrodinger equations, i jiaquan liu and zhiqiang wang communicated by david s. The nonlinearity here corresponds to the superfluid film equation in plasma physics. Uniqueness of the ground state solutions of quasilinear schrodinger. Bound states to critical quasilinear schrodinger equations. Chapter 4 schroedinger equation mit opencourseware. Pdf nonexistence of stable solutions for quasilinear schrodinger. Ground state solutions for a quasilinear schrodinger equation. Positive solutions for a quasilinear schr odinger equation. Many results on the existence of nontrivial solutions of 1. The first existence results for equations of the form of 1. Quasilinear schrodinger equations 3 in the theorem, wellposedness is taken to include the existence of a local solution, uniqueness, and continuous dependence on the initial datum.

The schrodinger equation the previous the chapters were all about kinematics how classical and relativistic particles, as well as waves, move in free space. Soliton solutions for quasilinear schrodinger equations. This paper is concerned with the quasilinear schrodinger equation 0. Existence of solutions to quasilinear schrodinger equations with indefinite potential zupei shen, zhiqing han abstract. The energy method 1 problems for lecture 1 10 lecture 2. In 2 alves and figueiredo extended this last result to the quasilinear. This search for an equation describing matter waves was carried out by erwin schroedinger. Pdf multiple solutions for quasilinear schrodinger equation. We study the existence and multiplicity of solutions for quasilinear elliptic equations of the formwhere, is the laplacian. The cauchy problem for the quasilinear schrodinger equation arxiv. Under suitable hypotheses, we obtain the existence of a least energy solution u. Existence of positive solutions for a quasilinear schrodinger equation. Pdf abstract in this paper, we study the nonexistence of stable solutions for the quasilinear schrodinger equation 0. Local time decay for a quasilinear schrodinger equation lin.

The cauchy problem for the quasilinear schrodinger. Pdf solutions for a quasilinear schrodinger equation. Newtons laws, the schrodinger equation does not give the trajectory of a particle, but rather the wave function of the quantum system, which carries information about the wave nature of the particle, which allows us to only discuss the probability of nding the particle in di erent regions of space. For example in 7, 22, 23,32333438 were considered this compactness condition in order to get existence and multiplicity of solutions for quasilinear schrodinger equations using the well. A parameterized quasilinear schrodinger equation with indefinite potentials. Rn, where g and v are periodic in x1,xnx1,xn and g is. See also special cases of the nonlinear schrodinger equation. Chapter 4 schroedinger equation einsteins relation between particle energy and frequency eq. Quasilinear equations such as have been accepted as models of several physical phenomena corresponding to various types of. Static solutions of a ddimensional modified nonlinear schrodinger equation, nonlinearity 16. However except when n 1 this functional is not defined on all h1rn. This work was supported by nsfc 116731 and nsffj 2014j06002. Therefore, the solution of the 3d schrodinger equation is obtained by multiplying the solutions of the three 1d schrodinger equations.

One can now substitute these expressions into the full 3d schrodinger equation and see that they solve it even at the points r where r 0. A quasilinear schrodinger equation for large amplitude. The schrodingers schrodingers equation is the basic equation of quantum mechanics. We begin by quoting the paper of cingolani and lazzo 8, which related the topology of the set of minima of v with the number of positive solutions of 1. Pdf bound states to critical quasilinear schrodinger. For a class of quasilinear schr odinger equations we establish the existence of ground states of soliton type solutions by. By a local linking theorem and the fountain theorem, we obtain the existence and multiplicity of solutions for the equation. In this article we study a quasilinear schr odinger equations with singularity. Ground state solutions for asymptotically periodic. On the existence of soliton solutions to quasilinear. We find the exact threshold depending upon the interplay of quasilinear and nonlinear terms that separates stability and instability. Pdf a parameterized quasilinear schr\odinger equation.

This handbook is intended to assist graduate students with qualifying examination preparation. We cite here some works which are closely related with our result. Received august 2017 revised april 2018 published august 2018. Nodal soliton solutions for generalized quasilinear. We consider quasilinear stationary schrodinger equations of the form 1uu2ugx,u,x. On the existence of soliton solutions to quasilinear schrodinger. We obtain a unique and positive solution by using the minimax method and some analysis techniques. Request pdf soliton solutions for quasilinear schrodinger equations, ii for a class of quasilinear schrodinger equations we establish the existence of ground states of soliton type solutions.

Multiple solutions for quasilinear schrodinger equation article pdf available in journal of differential equations 2544. Given the above considerations, it is natural to add some decay to the hs sobolev spaces where the quasilinear problem1. A parameterized quasilinear schrodinger equation with indefinite. The cauchy problem for the quasilinear schrodinger equation following kenigponcevega 1 lecture 1. Hunter department of mathematics, university of california at davis, usa and mihaela ifrim department of mathematics and statistics, mcmaster university, canada. Equation with positive coefficient in the quasilinear term and vanishing potential aires, jose f. In this paper, we consider the following quasilinear schrodinger equation. Existence and asymptotic profiles of positive solutions of. W e would like to point out that this work con tributes to the literature of quasilinear schr. Indefinite quasilinear elliptic equations in exterior domains with exponential critical growth alves, claudianor o. Nash moser methods for the solution of quasilinear schrodinger.