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Multiplicity of Positive Solutions for Weighted Quasilinear Elliptic Equations Involving Critical Hardy-Sobolev Exponents and Concave-Convex Nonlinearities
Tsing-San Hsu,Huei-Li Lin
Abstract and Applied Analysis , 2012, DOI: 10.1155/2012/579481
Abstract: By variational methods and some analysis techniques, the multiplicity of positive solutions is obtained for a class of weighted quasilinear elliptic equations with critical Hardy-Sobolev exponents and concave-convex nonlinearities.
Existence of solutions for quasilinear elliptic systems involving critical exponents and Hardy terms
Dengfeng Lu
Electronic Journal of Differential Equations , 2013,
Abstract: Using variational methods, including the Ljusternik-Schnirelmann theory, we prove the existence of solutions for quasilinear elliptic systems with critical Sobolev exponents and Hardy terms.
Existence and Multiplicity of Positive Solutions for a Quasilinear Elliptic Equation with Weighted Hardy-Sobolev Exponents

,SHANG Yan-ying

- , 2018, DOI: 10.13718/j.cnki.xdzk.2018.02.010
Abstract: 研究了一类加权拟线性椭圆方程,利用Ekeland变分原理和强极大值原理,证明了该方程正解的存在性和多重性.
We investigate a quasilinear elliptic equation with weighted Hardy-Sobolev exponents and, by means of Ekeland's variational principle and strong maximum principle, prove the existence and multiplicity of its positive solutions under different conditions
Interior gradient estimates for quasilinear elliptic equations  [PDF]
Truyen Nguyen,Tuoc Phan
Mathematics , 2015,
Abstract: We study quasilinear elliptic equations of the form $\text{div} \mathbf{A}(x,u,\nabla u) = \text{div}\mathbf{F} $ in bounded domains in $\mathbb{R}^n$, $n\geq 1$. The vector field $\mathbf{A}$ is allowed to be discontinuous in $x$, Lipschitz continuous in $u$ and its growth in the gradient variable is like the $p$-Laplace operator with $1p$, and we show that similar results also hold true in the setting of {\it Orlicz} spaces. Our regularity estimates extend results which are only known for the case $\mathbf{A}$ is independent of $u$ and they complement the well-known interior $C^{1,\alpha}$- estimates obtained by DiBenedetto \cite{D} and Tolksdorf \cite{T} for general quasilinear elliptic equations.
Estimates in the Hardy-Sobolev space of the annulus and stability result  [PDF]
Imed Feki
Mathematics , 2012,
Abstract: The main purpose of this work is to establish some logarithmic estimates of optimal type in the Hardy-Sobolev space $H^{k, \infty}; k \in {\mathbb{N}}^*$ of an annular domain. These results are considered as a continuation of a previous study in the setting of the unit disk by L. Baratchart and M. Zerner: On the recovery of functions from pointwise boundary values in a Hardy-sobolev class of the disk. J.Comput.Apll.Math 46(1993), 255-69 and by S. Chaabane and I. Feki: Logarithmic stability estimates in Hardy-Sobolev spaces $H^{k,\infty}$. C.R. Acad. Sci. Paris, Ser. I 347(2009), 1001-1006. As an application, we prove a logarithmic stability result for the inverse problem of identifying a Robin parameter on a part of the boundary of an annular domain starting from its behavior on the complementary boundary part.
Optimal logarithmic estimates in the Hardy-Sobolev space of the disk and stability results  [PDF]
Imed Feki,Houda Nfata,Franck Wielonsky
Mathematics , 2012,
Abstract: We prove a logarithmic estimate in the Hardy-Sobolev space $H^{k, 2}$, $k$ a positive integer, of the unit disk ${\mathbb D}$. This estimate extends those previously established by L. Baratchart and M. Zerner in $H^{1,2}$ and by S. Chaabane and I. Feki in $H^{k,\infty}$. We use it to derive logarithmic stability results for the inverse problem of identifying Robin's coefficients in corrosion detection by electrostatic boundary measurements and for a recovery interpolation scheme in the Hardy-Sobolev space $H^{k, 2}$ with interpolation points located on the boundary ${\mathbb T}$ of the unit disk.
Time-interior gradient estimates for quasilinear parabolic equations  [PDF]
Ben Andrews,Julie Clutterbuck
Mathematics , 2013,
Abstract: Bounded smooth solutions of the Dirichlet and Neumann problems for a wide variety of quasilinear parabolic equations, including graphical anisotropic mean curvature flows, have gradient bounded in terms of oscillation and elapsed time.
Asymptotic Behaviors of Solutions to quasilinear elliptic Equations with critical Sobolev growth and Hardy potential  [PDF]
Chang-Lin Xiang
Mathematics , 2015, DOI: 10.1016/j.jde.2015.05.007
Abstract: Optimal estimates on the asymptotic behaviors of weak solutions both at the origin and at the infinity are obtained to the following quasilinear elliptic equations \[-\Delta_{p}u-\frac{\mu}{|x|^{p}}|u|^{p-2}u=Q(x)|u|^{\frac{Np}{N-p}-2}u,\quad\,x\in \mathbb{R}^{N},\] where $1
Critical heat kernel estimates via Hardy-Sobolev inequalities  [PDF]
G. Barbatis,S. Filippas,A. Tertikas
Mathematics , 2003,
Abstract: We obtain Sobolev inequalities for the Schrodinger operator -\Delta-V, where V has critical behaviour V(x)=((N-2)/2)^2|x|^{-2} near the origin. We apply these inequalities to obtain pointwise estimates on the associated heat kernel, improving upon earlier results.
Lp estimates and asymptotic behavior for finite energy solutions of extremals to Hardy-Sobolev inequalities  [PDF]
Dimiter Vassilev
Mathematics , 2006,
Abstract: Motivated by the equation satisfied by the extremals of certain Hardy-Sobolev type inequalities, we show sharp $L^q$ regularity for finite energy solutions of p-laplace equations involving critical exponents and possible singularity on a sub-space of $\mathbb{R}^n$, which imply asymptotic behavior of the solutions at infinity. In addition, we find the best constant and extremals in the case of the considered $L^2$ Hardy-Sobolev inequality.
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