Abstract:
we consider the singular logarithmic potential , a potential which plays an important role in the modelling of triaxial systems, such as elliptical galaxies or bars in the centres of galaxy discs. using properties of the central field in the axis-symmetric case we obtain periodic solutions which are symmetric with respect to the origin for weak anisotropies. also we generalize our result in order to include more general perturbations of the logarithmic potential.

Abstract:
We consider the singular logarithmic potential , a potential which plays an important role in the modelling of triaxial systems, such as elliptical galaxies or bars in the centres of galaxy discs. Using properties of the central field in the axis-symmetric case we obtain periodic solutions which are symmetric with respect to the origin for weak anisotropies. Also we generalize our result in order to include more general perturbations of the logarithmic potential.

本文在Orlicz-Sobolev空间上给出了一种模范数，给出了由严格凸N函数生成Orlicz-Sobolev空间严格凸的充要条件。 In this paper we give a modular norm for Orlicz-Sobolev spaces, and obtain a necessary and suffi-cient condition for the Orlicz-Sobolev spaces which is formed by strictly convex N function to be rotund.

Abstract:
This article shows the existence of solutions to the nonlinear elliptic problem $A(u)=f$ in Orlicz-Sobolev spaces with a measure valued right-hand side, where $A(u)=-mathop{ m div}a(x,u, abla u)$ is a Leray-Lions operator defined on a subset of $W_{0}^{1}L_{M}(Omega)$, with general $M$.

Abstract:
本文研究了具光滑边界的有界域上拟线性椭圆问题的多解性.在Orlicz-Sobolev空间中利用变分及扰动的方法,得到了方程在对称及非对称情况下解的存在性和多解性. In this paper,we study multiplicity of solutions for the quasilinear elliptic problem in a bounded domain with smooth boundary.By using variational and perturbed methods in Orlicz-Sobolev space,we prove the existence of multiple solutions both in symmetric and nonsymmetric case

Abstract:
Let $Om$ be a bounded open set in $R^2$ sufficiently smooth and $f_k=(u_k,v_k)$ and $f=(u,v)$ mappings belong to the Sobolev space $W^{1,2}(Omega, R^2)$. We prove that if the sequence of Jacobians $J_{f_k}$ converges to a measure $mu$ in sense of measures and if one allows different assumptions on the two components of $f_k$ and $f$, e.g. $$ u_k ightharpoonup u ;;mbox{weakly in} ;; W^{1,2}(Omega) qquad , v_k ightharpoonup v ;;mbox{weakly in} ;; W^{1,q}(Omega) $$ for some $qin(1,2)$, then egin{equation}label{0} dmu=J_f,dz. end{equation} Moreover, we show that this result is optimal in the sense that conclusion fails for $q=1$. On the other hand, we prove that eqref{0} remains valid also if one considers the case $q=1$, but it is necessary to require that $u_k$ weakly converges to $u$ in a Zygmund-Sobolev space with a slightly higher degree of regularity than $W^{1,2}(Omega)$ and precisely $$ u_k ightharpoonup u ;;mbox{weakly in} ;; W^{1,L^2 log^alpha L}(Omega)$$ for some $alpha > 1$. Let $Om$ be a bounded open set in $R^2$ sufficiently smooth and $f_k=(u_k,v_k)$ and $f=(u,v)$ mappings belong to the Sobolev space $W^{1,2}(Om,R^2)$. We prove that if the sequence of Jacobians $J_{f_k}$ converges to a measure $mu$ in sense of measures and if one allows different assumptions on the two components of $f_k$ and $f$, e.g. $$ u_k ightharpoonup u ;;mbox{weakly in} ;; W^{1,2}(Om) qquad , v_k ightharpoonup v ;;mbox{weakly in} ;; W^{1,q}(Om) $$ for some $qin(1,2)$, then egin{equation}label{0} dmu=J_f,dz. end{equation} Moreover, we show that this result is optimal in the sense that conclusion fails for $q=1$. On the other hand, we prove that eqref{0} remains valid also if one considers the case $q=1$, but it is necessary to require that $u_k$ weakly converges to $u$ in a Zygmund-Sobolev space with a slightly higher degree of regularity than $W^{1,2}(Om)$ and precisely $$ u_k ightharpoonup u ;;mbox{weakly in} ;; W^{1,L^2 log^alpha L}(Om)$$ for some $alpha >1$.

Abstract:
We prove the existence of bounded solutions for the nonlinear elliptic problem$$ -mathop{m div}a(x,u,{ abla}u)=f quad ext{in }{Omega},$$ with $uin W^1_0L_M({Omega})cap L^{infty}(Omega)$, where$$ a(x,s,xi)cdotxigeq {overline M}^{-1}M(h(|s|))M(|xi|),$$ and $h:{mathbb{R}^+}{ o }{]0,1]}$ is a continuous monotone decreasing function with unbounded primitive. As regards the $N$-function $M$, no $Delta_2$-condition is needed.

Abstract:
Let $a:mathbb{R} o mathbb{R}$ be a strictly increasing odd continuous function with $lim_{t o +infty }a(t)=+infty $ and $A(t)=int_{0}^{t}a(s),ds$, $tin mathbb{R}$, the $N$-function generated by $a$. Let $Omega $ be a bounded open subset of $mathbb{R}^{N}$, $Ngeq 2$, $T[u,u]$ a nonnegative quadratic form involving the only generalized derivatives of order $m$ of the function $uin W_{0}^{m}E_{A}(Omega )$ and $g_{alpha }:Omega imesmathbb{R} omathbb{R}$, $| alpha | Keywords A priori estimate --- critical points --- Orlicz-Sobolev spaces --- Leray-Schauder topological degree --- Duality mapping --- Nemytskij operator --- Mountain Pass Theorem