共查询到20条相似文献,搜索用时 46 毫秒
1.
Let \(\Omega \subset \mathbb {R}^\nu \), \(\nu \ge 2\), be a \(C^{1,1}\) domain whose boundary \(\partial \Omega \) is either compact or behaves suitably at infinity. For \(p\in (1,\infty )\) and \(\alpha >0\), define where \(\mathrm {d}\sigma \) is the surface measure on \(\partial \Omega \). We show the asymptotics where \(H_\mathrm {max}\) is the maximum mean curvature of \(\partial \Omega \). The asymptotic behavior of the associated minimizers is discussed as well. The estimate is then applied to the study of the best constant in a boundary trace theorem for expanding domains, to the norm estimate for extension operators and to related isoperimetric inequalities.
相似文献
$$\begin{aligned} \Lambda (\Omega ,p,\alpha ):=\inf _{\begin{array}{c} u\in W^{1,p}(\Omega )\\ u\not \equiv 0 \end{array}}\dfrac{\displaystyle \int _\Omega |\nabla u|^p \mathrm {d} x - \alpha \displaystyle \int _{\partial \Omega } |u|^p\mathrm {d}\sigma }{\displaystyle \int _\Omega |u|^p\mathrm {d} x}, \end{aligned}$$
$$\begin{aligned} \Lambda (\Omega ,p,\alpha )=-(p-1)\alpha ^{\frac{p}{p-1}} - (\nu -1)H_\mathrm {max}\, \alpha + o(\alpha ), \quad \alpha \rightarrow +\infty , \end{aligned}$$
2.
Jacques Giacomoni Sweta Tiwari Guillaume Warnault 《NoDEA : Nonlinear Differential Equations and Applications》2016,23(3):24
We discuss the existence and uniqueness of the weak solution of the following quasilinear parabolic equationinvolving the p(x)-laplacian operator. Next, we discuss the global behaviour of solutions and in particular some stabilization properties.
相似文献
$$\left\{\begin{array}{ll}u_t-\Delta _{p(x)}u = f(x,u)&\quad \text{in }\quad Q_T \stackrel{{\rm{def}}}{=} (0,T)\times\Omega,\\u = 0 & \quad\text{on}\quad \Sigma_T\stackrel{{\rm{def}}}{=} (0,T)\times\partial\Omega,\\u(0,x)=u_0(x)& \quad \text{in}\quad \Omega \end{array}\right.\quad\quad (P_{T})$$
3.
César E. Torres Ledesma Nemat Nyamoradi 《Journal of Applied Mathematics and Computing》2017,55(1-2):257-278
In the present paper, we deal with the existence and multiplicity of solutions for the following impulsive fractional boundary value problem where \(\alpha \in (1/p, 1]\), \(1<p<\infty \), \(0 = t_0<t_1< t_2< \cdots< t_n < t_{n+1} = T\), \(f:[0,T]\times \mathbb {R} \rightarrow \mathbb {R}\) and \(I_j : \mathbb {R} \rightarrow \mathbb {R}\), \(j = 1, \ldots , n\), are continuous functions, \(a\in C[0,T]\) and By using variational methods and critical point theory, we give some criteria to guarantee that the above-mentioned impulsive problems have at least one weak solution and a sequences of weak solutions.
相似文献
$$\begin{aligned} {_{t}}D_{T}^{\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t)\right) + a(t)|u(t)|^{p-2}u(t)= & {} f(t,u(t)),\;\;t\ne t_j,\;\;\hbox {a.e.}\;\;t\in [0,T],\\ \Delta \left( {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t_j)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t_j)\right) \right)= & {} I_j(u(t_j))\;\;j=1,2,\ldots ,n,\\ u(0)= & {} u(T) = 0. \end{aligned}$$
$$\begin{aligned} \Delta \left( {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t_j)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t_j)\right) \right)= & {} {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u\left( t_j^+\right) \right| ^{p-2}{_{0}}D_{t}^{\alpha }u\left( t_j^+\right) \right) \\&- {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t_j^-)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u\left( t_j^-\right) \right) ,\\ {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u\left( t_j^+\right) \right| ^{p-2}{_{0}}D_{t}^{\alpha }u\left( t_j^+\right) \right)= & {} \lim _{t \rightarrow t_j^+} {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t)\right) ,\\ {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t_j^-)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t_j^-)\right)= & {} \lim _{t\rightarrow t_j^-}{_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t)\right) . \end{aligned}$$
4.
This paper is concerned with the blow-up of solutions to the following nonlocal p-Laplace equation:under homogeneous Neumann boundary conditions in a bounded smooth domain \({\Omega\subset\mathrm{R}^N}\). For all \({p > 2, q > p-1}\), a blow-up result for the solutions to the above equation with positive initial energy is established. This result improves a recent result by Qu and Liang (Abstr Appl Anal 3:551–552, 2013) which asserts the blow-up of solutions for \({p-1 < q\leq\frac{Np}{(N-p)_+}-1}\).
相似文献
$$u_t-\mathrm{div}(|\nabla{u}|^{p-2}\nabla{u})=|u|^{q-1}u-\frac{1}{|\Omega|} \int\limits_\Omega{|u|^{q-1}u}dx,\quad x\in\Omega,\quad 0 < t < T,$$
5.
Fengping Yao 《Archiv der Mathematik》2017,108(1):85-97
In this paper we give a new alternative proof of the local higher integrability in Orlicz spaces of the gradient for weak solutions of quasilinear parabolic equations of p-Laplacian typefor any p > 0. Moreover, we point out that our results are homogeneousregularity estimates in Orlicz spaces and improve the known results for such equations by using some new techniques. Actually, our results can be extended to the global estimates and cover a more general class of degenerate/singular parabolic problems of p-Laplacian type.
相似文献
$$\begin{array}{ll} u_t-\text{div} \left( \left | \nabla u\right|^{ p-2 } \nablau\right)=\text{div} \left(| \mathrm{ \bf f}|^{p-2} \mathrm{ \bf f}\right)\quad {\rm in}~\Omega\times (0,T] \end{array}$$
6.
In this paper, we study the existence result for the nonlinear fractional differential equations with p-Laplacian operatorwhere the p-Laplacian operator is defined as \({\phi_p(s) = |s|^{p-2}s,p > 1, \,\,{\rm and}\,\, \phi_q(s) = \phi_p^{-1}(s), \frac{1}{p}+\frac{1}{q} = 1;\, 0 < \alpha, \beta < 1, 1 < \alpha + \beta < 2 \,\,{\rm and}\,\, D_{0^+}^{\alpha}, D_{0^+}^{\beta}}\) denote the Caputo fractional derivatives, and \({f : [0,1] \times \mathbb{R}^2\rightarrow \mathbb{R}}\) is continuous. Though Chen et al. have studied the same equations in their article, the proof process is not rigorous. We point out the mistakes and give a correct proof of the existence result. The innovation of this article is that we introduce a new definition to weaken the conditions of Arzela–Ascoli theorem and overcome the difficulties of the proof of compactness of the projector K P (I ? Q)N. As applications, an example is presented to illustrate the main results.
相似文献
$$\left\{\begin{array}{ll}D_{0^+}^{\beta} \phi_p( D_{0^+}^{\alpha} u(t))=f(t,u(t),D_{0^+}^{\alpha}u(t)), \quad t\in(0,1),\\ D_{0^+}^{\alpha}u(0)=D_{0^+}^{\alpha}u(1)=0,\end{array}\right.$$
7.
A remark on the existence of entire large and bounded solutions to a (<Emphasis Type="Italic">k</Emphasis><Subscript>1</Subscript>, <Emphasis Type="Italic">k</Emphasis><Subscript>2</Subscript>)-Hessian system with gradient term
下载免费PDF全文
![点击此处可从《数学学报(英文版)》网站下载免费的PDF全文](/ch/ext_images/free.gif)
Dragos Patru Covei 《数学学报(英文版)》2017,33(6):761-774
In this paper, we study the existence of positive entire large and bounded radial positive solutions for the following nonlinear system Here \({S_{{k_i}}}\left( {\lambda \left( {{D^2}{u_i}} \right)} \right)\) is the k i -Hessian operator, a 1, p 1, f 1, a 2, p 2 and f 2 are continuous functions.
相似文献
$$\left\{ {\begin{array}{*{20}c}{S_{k_1 } \left( {\lambda \left( {D^2 u_1 } \right)} \right) + a_1 \left( {\left| x \right|} \right)\left| {\nabla u_1 } \right|^{k_1 } = p_1 \left( {\left| x \right|} \right)f_1 \left( {u_2 } \right)} & {for x \in \mathbb{R}^N ,} \\{S_{k_2 } \left( {\lambda \left( {D^2 u_2 } \right)} \right) + a_2 \left( {\left| x \right|} \right)\left| {\nabla u_2 } \right|^{k_2 } = p_2 \left( {\left| x \right|} \right)f_2 \left( {u_1 } \right)} & {for x \in \mathbb{R}^N .} \\\end{array} } \right.$$
8.
We establish multiplicity and nonexistence of solutions to the quasilinear problem in some bounded smooth domains \(\Omega \) in \(\mathbb {R}^{N}\), for \(1<p<N\) and some supercritical exponents \(q>p^{*}:=\frac{Np}{N-p}\). Multiplicity is established in domains arising from the Hopf maps. We show that, after a suitable change of metric, these maps become p-harmonic morphisms, i.e., they preserve the p-Laplace operator up to a factor. We use them to reduce the supercritical problem to an anisotropic quasilinear critical problem in a domain of lower dimension.
相似文献
$$\begin{aligned} -\Delta _{p}v=\left| v\right| ^{q-2}v\,\,\text {in}\,\,\Omega ,\qquad v=0\text { on }{\partial {\Omega }}, \end{aligned}$$
9.
This paper deals with the existence and stability properties of positive weak solutions to classes of nonlinear systems involving
the (p,q)-Laplacian of the form
$ \left\{{ll} -\Delta_{p} u = \lambda \,a(x)\,v^{\alpha}-c, & x\in \Omega,\\ -\Delta_{q} v = \lambda \,b(x)\,u^{\beta}-c, & x\in \Omega,\\ u=0=v, & x\in\partial \Omega, \right. $ \left\{\begin{array}{ll} -\Delta_{p} u = \lambda \,a(x)\,v^{\alpha}-c, & x\in \Omega,\\ -\Delta_{q} v = \lambda \,b(x)\,u^{\beta}-c, & x\in \Omega,\\ u=0=v, & x\in\partial \Omega, \end{array}\right. 相似文献
10.
We study both the existence and uniqueness of nonnegative solution to a singular elliptic problem of Kirchhoff type, whose model is: 相似文献
$$\begin{aligned} {\left\{ \begin{array}{ll} -B\left( \dfrac{1}{2}\displaystyle \int _\Omega |\nabla u|^2\mathrm {d}x\right) \Delta u=\dfrac{h(x)}{u^\gamma }, &{}\quad x\in \Omega ,\\ u>0, &{}\quad x\in \Omega ,\\ u=0, &{}\quad x\in \partial \Omega , \end{array}\right. } \end{aligned}$$ 11.
We derive global gradient estimates for \(W^{1,p}_0(\Omega )\)-weak solutions to quasilinear elliptic equations of the form 相似文献
$$\begin{aligned} \mathrm {div\,}\mathbf {a}(x,u,Du)=\mathrm {div\,}(|F|^{p-2}F) \end{aligned}$$ 12.
Study the following K-component elliptic system Here \(k\ge 2\) is a integer and \(\Omega \subset \mathbb {R}^N(N\ge 4)\) is a bounded domain with smooth boundary \(\partial \Omega \), \(a_i,\lambda _i>0\), \(b_i\ge 0\) for all \(i=1,2,\ldots ,k\) and \(\beta <0\), \(2^*=\frac{2N}{N-2}\) is the Sobolev critical exponent. By the variational method, we obtain a nontrivial solution of this system. The concentration behavior of this nontrivial solution as \(\overrightarrow{\mathbf {b}}\rightarrow \overrightarrow{\mathbf {0}}\) and \(\beta \rightarrow -\infty \) are both studied and the phase separation is exhibited for \(N\ge 6\), where \(\overrightarrow{\mathbf {b}}=(b_1,b_2,\ldots ,b_k)\) is a vector. Our results extend and generalize the results in Chen and Zou (Arch Ration Mech Anal 205:515–551, 2012; Calc Var Partial Differ Equ 52:423–467, 2015). Moreover, by studying the phase separation, we also prove some existence and multiplicity results of the sign-changing solutions to the following Brezís–Nirenberg problem of the Kirchhoff type 相似文献
$$\begin{aligned} \left\{ \begin{array}{ll} -\bigg (a+b\int _{\Omega }|\nabla u|^2dx\bigg )\Delta u = \lambda u +|u|^{2^*-2}u, &{}\quad \text {in }\Omega , \\ u =0,&{}\quad \text {on }\partial \Omega , \end{array} \right. \end{aligned}$$ 13.
We prove that, for all integers \(n\ge 1\), 相似文献
$$\begin{aligned} \Big (\sqrt{2\pi n}\Big )^{\frac{1}{n(n+1)}}\left( 1-\frac{1}{n+a}\right) <\frac{\root n \of {n!}}{\root n+1 \of {(n+1)!}}\le \Big (\sqrt{2\pi n}\Big )^{\frac{1}{n(n+1)}}\left( 1-\frac{1}{n+b}\right) \end{aligned}$$ $$\begin{aligned} \big (\sqrt{2\pi n}\big )^{1/n}\left( 1-\frac{1}{2n+\alpha }\right) <\left( 1+\frac{1}{n}\right) ^{n}\frac{\root n \of {n!}}{n}\le \big (\sqrt{2\pi n}\big )^{1/n}\left( 1-\frac{1}{2n+\beta }\right) , \end{aligned}$$ $$\begin{aligned}&a=\frac{1}{2},\quad b=\frac{1}{2^{3/4}\pi ^{1/4}-1}=0.807\ldots ,\quad \alpha =\frac{13}{6} \\&\text {and}\quad \beta =\frac{2\sqrt{2}-\sqrt{\pi }}{\sqrt{\pi }-\sqrt{2}}=2.947\ldots . \end{aligned}$$ 14.
The existence of infinitely many solutions of the following Dirichlet problem for p-mean curvature operator:
is considered, where Θ is a bounded domain in R
n
(n>p>1) with smooth boundary ∂Θ. Under some natural conditions together with some conditions weaker than (AR) condition, we prove that the above problem
has infinitely many solutions by a symmetric version of the Mountain Pass Theorem if
.
Supported by the National Natural Science Foundation of China (10171032) and the Guangdong Provincial Natural Science Foundation
(011606). 相似文献
15.
We calculate the sharp bounds for some q-analysis variants of Hausdorff type inequalities of the form 相似文献
$$\int_0^{ + \infty } {{{\left( {\int_0^{ + \infty } {\frac{{\phi \left( t \right)}}{t}f\left( {\frac{x}{t}} \right){d_q}t} } \right)}^p}{d_q}x} \leqslant {C_\phi }\int_0^b {{f^p}\left( t \right)} {d_q}t$$ 16.
Huixue Lao 《Acta Appl Math》2010,110(3):1127-1136
Let L(sym j f,s) be the jth symmetric power L-function attached to a holomorphic Hecke eigencuspform f(z) for the full modular group, and \(\lambda_{\mathrm{sym}^{j}f}(n)\) denote its nth coefficient. In this paper we are able to prove that 相似文献
$\int_{1}^{x}\bigg|\sum_{n\leq y}\lambda_{\mathrm{sym}^{3}f}(n)\bigg|^{2}dy=O\bigl(x^{2}\bigr),$ $\int_{1}^{x}\bigg|\sum_{n\leq y}\lambda_{\mathrm{sym}^{4}f}(n)\bigg|^{2}dy=O\bigl(x^{\frac{11}{5}}\log x\bigr).$ 17.
Let M Ω be the maximal operator with homogeneous kernel Ω. In the present paper, we show that if Ω satisfies the L 1-Dini condition on ?? n?1, then the following weak type (1,1) behaviors hold for the maximal operator M Ω and \(f\in L^{1}(\mathbb {R}^{n})\), here \(\tilde {\omega }_{1}\) denotes the L 1 integral modulus of continuity of Ω defined by translation in \(\mathbb {R}^{n}\). 相似文献
$$\lim\limits _{\lambda \rightarrow 0_{+}}\lambda m(\{x\in \mathbb {R}^{n}:M_{\Omega } f(x)>\lambda \})=\frac {1}{n} \|\Omega \|_{1} \|f\|_{1},$$ $$\sup\limits_{\lambda >0}\lambda m(\{x\in \mathbb {R}^{n}:M_{\Omega } f(x)>\lambda \})\lesssim {\bigg ((\log n)\|\Omega \|_{1}+{\int }_{0}^{1/n}\frac {\tilde {\omega }_{1}(\delta )}{\delta }d\delta \bigg )}\|f\|_{1}$$ 18.
Lucio Boccardo 《Milan Journal of Mathematics》2011,79(1):193-206
The aim of this work is to study the existence of solutions of quasilinear elliptic problems of the type
|
设为首页 | 免责声明 | 关于勤云 | 加入收藏 |
Copyright©北京勤云科技发展有限公司 京ICP备09084417号 |