MULTIPLE AND SIGN-CHANGING SOLUTIONS FOR NONLINEAR ELLIPTIC EQUATION WITH CRITICAL POTENTIAL AND CRITICAL PARAMETER

Some embedding inequalities in Hardy-Sobolev space are proved. Furthermore, by the improved inequalities and the linking theorem, in a new k-order SobolevHardy space, we obtain the existence of sign-changing solutions for the nonlinear elliptic equation {-△（k）u：=-△u-（N-2）2/4u/｜x｜2-1/4k-1∑im1u/｜x｜2（ln（i）R/｜x｜2=f（x,u）,x∈Ω,u=0,x∈Ω,where 0∈ΩBa（0）RN,n≥3,ln）i）=6jm1ln（j）,and R=ae（k-1）,where e（0）=1,e（j）=ee（j=1）for j≥1,ln（1）=ln,ln（j）=lnln（j-1）for j≥2.Besides,positive and negative solutions are obtained by a variant mountain pass theorem.

Nonlinear Biharmonic Equations with Critical Potential

In this paper, we study two semilinear singular biharmonic equations： one with subcritical exponent and critical potential, another with sub-critical potential and critical exponent. By Pohozaev identity for singular solution, we prove there is no nontrivial solution for equations with critical exponent and critical potential. And by using the concentrate compactness principle and Mountain Pass theorem, respectively, we get two existence results for the two problems. Meanwhile, we have compared the changes of the critical dimensions in singular and non-singular cases, and we get an interesting result.

Singular Quasilinear Elliptic Problems with Indefinite Weights and Critical Potential

The Dirichlet problem for a quasilinear sub-critical inhomogeneous elliptic equation with critical potential and singular coefficients, which has indefinite weights in RN, is studied in this paper. We discuss the corresponding eigenvalue problems by the variational techniques and Picone＇s identity, and obtain the existence of non-trivial solutions for the inhomogeneous Dirichlet problem by using Hardy inequality, Mountain Pass Lemma in conjunction with the property of eigenvalues.

EXISTENCE RESULTS FOR DEGENERATE ELLIPTIC EQUATIONS WITH CRITICAL CONE SOBOLEV EXPONENTS

In this paper, we study the existence result for degenerate elliptic equations with singular potential and critical cone sobolev exponents on singular manifolds. With the help of the variational method and the theory of genus, we obtain several results under different conditions.

THE EXISTENCE AND LOCAL UNIQUENESS OF MULTI-PEAK SOLUTIONS TO A CLASS OF KIRCHHOFF TYPE EQUATIONS

In this paper,we study the existence and local uniqueness of multi-peak solutions to the Kirchhoff type equations-(ε^(2)a+εb∫_(R^(3))|■u|^(2))△u+V(x)u=u^(p),u>0 in R^(3),which concentrate at non-degenerate critical points of the potential function V(x),where a,b>0,1<p<5 are constants,andε>0 is a parameter.Applying the Lyapunov-Schmidt reduction method and a local Pohozaev type identity,we establish the existence and local uniqueness results of multi-peak solutions,which concentrate at{a_(i)}1≤i≤k,where{a_(i)}1≤i≤k are non-degenerate critical points of V(x)asε→0.

Vibration and Buckling Analysis of Piezoelectric Nanowires Based on Surface Energy Density

In this paper,the influences of surface effects on free transverse vibration and buckling of piezoelectric nanowires are investigated by surface energy density elasticity theory.Analytical relations are given for the natural frequencies of nanowires by taking into account the effects of surface energy density and surface relaxation parameter.By implementing this theory with consideration of surface effects under clamped-clamped boundary conditions,the natural frequencies of nanowires are calculated.It is shown that the natural frequency depends on both the surface effects and piezoelectricity.A closed-form solution is also obtained to calculate the critical buckling voltage.This study is expected to provide useful insights for the design of piezoelectric nanowire-based nanodevices.