Schrodinger equation with a cubic nonlinearity, schrodinger equation with a powerlaw nonlinearity. 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. We consider quasilinear stationary schrodinger equations of the form 1uu2ugx,u,x. On the existence of soliton solutions to quasilinear schrodinger. In this article we study a quasilinear schr odinger equations with singularity. Introduction and statement of main results this article concerns the singular quasilinear schrodinger equation with the dirichlet boundary value condition. 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.
This handbook is intended to assist graduate students with qualifying examination preparation. 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. The cauchy problem for the quasilinear schrodinger equation following kenigponcevega 1 lecture 1. Pdf a parameterized quasilinear schr\odinger equation. Chapter 4 schroedinger equation einsteins relation between particle energy and frequency eq. The description of nature is essentially probabilistic, with the probability of an. Stability of standing waves for a class of quasilinear. Ground state solutions for asymptotically periodic.
Static solutions of a ddimensional modified nonlinear schrodinger equation, nonlinearity 16. Nodal soliton solutions for generalized quasilinear. Pdf bound states to critical quasilinear schrodinger. Equation with positive coefficient in the quasilinear term and vanishing potential aires, jose f. Ground state solutions for a quasilinear schrodinger equation. The energy method 1 problems for lecture 1 10 lecture 2. Pdf multiple solutions for quasilinear schrodinger equation. We find the exact threshold depending upon the interplay of quasilinear and nonlinear terms that separates stability and instability. Indefinite quasilinear elliptic equations in exterior domains with exponential critical growth alves, claudianor o. Rn, where g and v are periodic in x1,xnx1,xn and g is. Soliton solutions for quasilinear schrodinger equations, i jiaquan liu and zhiqiang wang communicated by david s. Under suitable hypotheses, we obtain the existence of a least energy solution u. The schrodinger equation the previous the chapters were all about kinematics how classical and relativistic particles, as well as waves, move in free space. 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.
The cauchy problem for the quasilinear schrodinger equation arxiv. A parameterized quasilinear schrodinger equation with indefinite. Existence of solutions to quasilinear schrodinger equations with indefinite potential zupei shen, zhiqing han abstract. Pdf nonexistence of stable solutions for quasilinear schrodinger. W e would like to point out that this work con tributes to the literature of quasilinear schr. 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. Existence and asymptotic profiles of positive solutions of. Positive solutions for a quasilinear schr odinger equation. Quasilinear equations such as have been accepted as models of several physical phenomena corresponding to various types of. Existence of positive solutions for a quasilinear schrodinger equation. A quasilinear schrodinger equation for large amplitude. The first existence results for equations of the form of 1. Soliton solutions for quasilinear schrodinger equations. Pdf solutions for a quasilinear schrodinger equation.
Many results on the existence of nontrivial solutions of 1. Such equations have been derived as models of several physical phenomena. Uniqueness of the ground state solutions of quasilinear schrodinger. This search for an equation describing matter waves was carried out by erwin schroedinger. Nash moser methods for the solution of quasilinear schrodinger. This paper is concerned with the quasilinear schrodinger equation 0. Existence and concentration of positive solutions for.
We achieved our results by using minimax methods and lusternikschnirelman theory of critical points. Chapter 4 schroedinger equation mit opencourseware. In fact, this particular case will cover most of the problems that well encounter in ee 439. The above theorem also holds for ultrahyperbolic operators, as in 5, 6, where g0 is of a di erent signature. We study the existence and multiplicity of solutions for quasilinear elliptic equations of the formwhere, is the laplacian. The schrodingers schrodingers equation is the basic equation of quantum mechanics.
Local time decay for a quasilinear schrodinger equation lin. In 2 alves and figueiredo extended this last result to the quasilinear. A parameterized quasilinear schrodinger equation with indefinite potentials. See also special cases of the nonlinear schrodinger equation.
The cauchy problem for the quasilinear schrodinger. In this paper, we study the existence and the properties of standing waves of the form u. Therefore, the solution of the 3d schrodinger equation is obtained by multiplying the solutions of the three 1d schrodinger equations. Soliton solutions for quasilinear schrodinger equations, i. Given the above considerations, it is natural to add some decay to the hs sobolev spaces where the quasilinear problem1. The nonlinearity here corresponds to the superfluid film equation in plasma physics. However except when n 1 this functional is not defined on all h1rn. Introducing a change of unknown, we transform the search of solutions ux of 1 into the search of solutions vx of the semilinear equation. Multiple solutions for quasilinear schrodinger equation article pdf available in journal of differential equations 2544. This work was supported by nsfc 116731 and nsffj 2014j06002. Hunter department of mathematics, university of california at davis, usa and mihaela ifrim department of mathematics and statistics, mcmaster university, canada. 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.
We cite here some works which are closely related with our result. 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. Received august 2017 revised april 2018 published august 2018. We obtain a unique and positive solution by using the minimax method and some analysis techniques. Quasilinear schrodinger equation, ljusternikschnirelmann theory, positive solutions, critical problems. Multiple positive solutions for a quasilinear system of schrodinger. On the existence of soliton solutions to quasilinear.
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