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1.
Hardy type and Rellich type inequalities on the Heisenberg group   总被引:13,自引:0,他引:13  

This paper contains some interesting Hardy type inequalities and Rellich type inequalities for the left invariant vector fields on the Heisenberg group.

  相似文献

2.
Multiple solutions of some boundary value problems with parameters   总被引:1,自引:0,他引:1  
In this paper, we study the existence and multiplicity of nontrivial solutions for the following second-order Dirichlet nonlinear boundary value problem with odd order derivative: −u(t)+au(t)+bu(t)=f(t,u(t)) for all t∈[0,1] with u(0)=u(1)=0, where a,bR1, fC1([0,1]×R1,R1). By using the Morse theory, we impose certain conditions on f which are able to guarantee that the problem has at least one nontrivial solution, two nontrivial solutions and infinitely many solutions, separately.  相似文献
3.
This paper is concerned with partial regularity for weak solutions to nonlinear sub-elliptic systems in divergence form in Carnot groups. The technique of A-harmonic approximation introduced by Simon and developed by Duzaar and Grotowski is adapted to our context. We establish Caccioppoli type inequalities and partial regularity with optimal local Hölder exponents for horizontal gradients of weak solutions to systems under super-quadratic natural structure conditions and super-quadratic controllable structure conditions, respectively.  相似文献
4.
Hardy-Sobolev type inequalities on the H-type group   总被引:1,自引:0,他引:1  
Motivated by the idea of Badiale and Tarantello who have found Hardy-Sobolev inequalities on Rn, a class of Hardy-Sobolev type inequalities on H-type groups is proved via a new representation formula for functions. Extremal functions realizing equality in the inequalities are discussed by refined Concentration-Compactness principles. Finally, some sharp constants for Hardy type inequalities are given. The project supported by National Natural Science Foundation of China, Grant No. 10371099.  相似文献
5.
Let α and s be real numbers satisfying 0<s<α<n. We are concerned with the integral equation $$u(x)=\int_{R^n}\frac{u^p(y)}{|x-y|^{n-\alpha}|y|^s}dy, $$ where \(\frac{n-s}{n-\alpha}< p< \alpha^{*}(s)-1\) with \(\alpha^{*}(s)=\frac{2(n-s)}{n-\alpha}\) . We prove the nonexistence of positive solutions for the equation and establish the equivalence between the above integral equation and the following partial differential equation $$\begin{aligned} (-\Delta)^{\frac{\alpha}{2}}u(x)=|x|^{-s}u^p. \end{aligned}$$   相似文献
6.
In this paper, by investigating the effect of the subcritical terms and the coefficients of the singular terms, some existence results for quasilinear elliptic problems involving combined critical Sobolev–Hardy terms are obtained via variational methods.  相似文献
7.
We consider a class of Kolmogorov equation $$Lu={\sum^{p_0}_{i,j=1}{\partial_{x_i}}(a_{ij}(z){\partial_{x_j}}u)}+{\sum^{N}_{i,j=1}b_{ij}x_{i}{\partial_{x_j}}u-{\partial_t}u}={\sum^{p_0}_{j=1}{\partial_{x_j}}F_{j}(z)}$$ in a bounded open domain ${\Omega \subset \mathbb{R}^{N+1}}$ , where the coefficients matrix (a ij (z)) is symmetric uniformly positive definite on ${\mathbb{R}^{p_0} (1 \leq p_0 < N)}$ . We obtain interior W 1,p (1 < p < ∞) regularity and Hölder continuity of weak solutions to the equation under the assumption that coefficients a ij (z) belong to the ${VMO_L\cap L^\infty}$ and ${({b_{ij}})_{N \times N}}$ is a constant matrix such that the frozen operator ${L_{z_0}}$ is hypoelliptic.  相似文献
8.
Let G be a homogeneous group, and let X 1, X 2, … , X m be left invariant real vector fields being homogeneous of degree one on G. We consider the following Dirichlet boundary value problem of the sub-Laplace equation involving the critical exponent and singular term: $$\left\{\begin{array}{ll}-\sum_{j=1}^{m}X_j^2u(x)-\frac{a}{\|x\|^\nu}u(x)=u^{\frac{Q+2}{Q-2}}(x), x\in\Omega,\\ u(x)=0, \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\,\,\,\, x\in \partial\Omega,\end{array}\right.$$ where ${\Omega\subset G}$ is a bounded domain with smooth boundary and ${\mathbf{0}\in\Omega}$ , Q is the homogeneous dimension of G, ${a\in \mathbb{R},\ \nu <2 }$ . We boost u to ${L^p(\Omega)}$ for any ${1\leq p < \infty}$ if ${u\in S^{1,2}_0(\Omega)}$ is a weak solution of the problem above.  相似文献
9.
In this paper, we establish gradient estimates in Morrey spaces and H?lder continuity for weak solutions of the following degenerate elliptic system $$-X_{\alpha}^{\ast}(a_{ij}^{\alpha\beta}(x)X_{\beta}u^{j})=g_{i}-X_{\alpha}^{\ast}f_{i}^{\alpha}(x),$$ where X 1, . . . , X q are real smooth vector fields satisfying H?rmander’s condition, coefficients ${a_{ij}^{\alpha \beta }\in VMO_X \cap L^\infty (\Omega ), \alpha,\beta=1,2, \,.\,.\,.\, ,q, i,j=1,2, \,.\,.\,.\, ,N, X_{\alpha}^{\ast}}$ is the transposed vector field of X α.  相似文献
10.
In this paper, we consider the elliptic equations with critical Sobolev exponents and multi-polar potentials in bounded symmetric domains and prove the existence and multiplicity of symmetric positive solutions by using the Ekeland variational principle and the Lusternik–Schnirelmann category theory.  相似文献
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