EXISTENCE OF NONTRIVIAL SOLUTIONS FOR GENERALIZED QUASILINEAR SCHRDINGER EQUATIONS WITH CRITICAL OR SUPERCRITICAL GROWTHS

In this paper, we study the following generalized quasilinear Schrodinger equa- tions with critical or supercritical growths-div（g2（u）△u） ＋ g（u）g＇（u）｜△u｜2 ＋ V（x）u = f（x, u） ＋ λ｜u｜P-2 u, x ∈ RN,where λ 〉 0, N ≥ 3, g ： R →R＋ is a C1 even function, g（0） = 1, g＇（s） ≥ 0 for all s ≥ 0, lim ｜s｜→＋ ∞g（s）/｜s｜α-1：= β 〉 0 for some α≥ 1 and （α- 1）g（s） 〉 g＇（s）s for all s 〉 0 and p≥α2＊.Under some suitable conditions, we prove that the equation has a nontrivial solution for smallλ 〉 0 using a change of variables and variational method.

MULTIPLE NONTRIVIAL SOLUTIONS FOR A CLASS OF SEMILINEAR POLYHARMONIC EQUATIONS

In this paper, we are concerned with the following problem：{（-△）ku=λf（x）｜u｜q-2u＋g（x）｜u｜k＊-2u, x∈Ω, u∈H k0 （Ω）, where Ωis a bounded domain in RN with N ≥2k＋1, 1〈q〈2,λ〉0, f, g are continuous functions on Ω which are somewhere positive but which may change sign on Ω. k＊ = N2/N-2k is the critical Sobolev exponent. By extracting the Palais-Smale sequence in the Nehari manifold, the existence of multiple nontrivial solutions to this equation is verified.

Ground states for asymptotically periodic quasilinear Schrdinger equations with critical growth

For a class of asymptotically periodic quasilinear Schr？dinger equations with critical growth the existence of ground states is proved.First applying a change of variables the quasilinear Schr？dinger equations are reduced to semilinear Schr？dinger equations in which the corresponding functional is well defined in H1 RN .Moreover there is a one-to-one correspondence between ground states of the semilinear Schr？dinger equations and the quasilinear Schr？dinger equations.Then the mountain-pass theorem is used to find nontrivial solutions for the semilinear Schr？dinger equations. Finally under certain monotonicity conditions using the Nehari manifold method and the concentration compactness principle the nontrivial solutions are found to be exactly the same as the ground states of the semilinear Schr？dinger equations.

Constraint-induced restriction and extension operators with applications

The Stokes operator is a differential-integral operator induced by the Stokes equations. In this paper, we analyze the Stokes operator from the point of view of the Helmholtz minimum dissipation principle. We show that, through the Hodge orthogonal decomposition, a pair of bounded linear operators, a restriction operator and an extension operator, are induced from the divergence-free constraint. As a consequence, we use it to calculate the eigenvalues of the Stokes operator.

Study on Retrieving Three-Dimensional Wind Using Simulated Dual-Doppler Radar Data in the Cartesian Space

This paper investigates a technique of retrieving three-dimensional windfields from the dual-Doppler weather radar radial wind which is based on the Cartesian space usingvariational method. This technology provides a simultaneous resolution of three wind components andsatisfies both the minimal dual-equation system and the continuity equation. The main advantage ofthis method is that it can remove the potential drawback of an iterative solution of Cartesiandual-Doppler analysis techniques which is a major demerit when one retrieves the vertical velocityusing mass continuity equation with iterative method. The data pre-processing technology andinterpolation are also studied. This work developed a three-dimensional Cressman weighting functionto process the interpolation. In order to test the capability and advantage of this method, onenumerical experiment based on simulating dual-Doppler radar observations is designed. Firstly, wesynthesize the dual-Doppler radar radial velocity and reflectivity from the numerical model. Then,the three-dimensional wind components are retrieved from the radial velocity and reflectivity usingthis technique. The retrieved three-dimensional wind fields are found to be quite consisted withthose previously simulated wind fields. Mean difference, root-mean-square error, and relativedeviation are defined to test the precision of the method. These statistic errors reveal theaccuracy and the advantage of this method. The numerical experiment has definitely testified thatthis technique can be used to retrieve the three-dimensional wind fields from the radial velocityand reflectivity detected by the real dual-Doppler weather radar.