共查询到20条相似文献,搜索用时 15 毫秒
1.
Yan Sheng SHEN 《数学学报(英文版)》2023,(11):2181-2206
In this paper we study the existence of nontrivial solutions to the well-known Brezis–Nirenberg problem involving the fractional p-Laplace operator in unbounded cylinder type domains.By means of the fractional Poincaré inequality in unbounded cylindrical domains, we first study the asymptotic property of the first eigenvalue λp,s(■) with respect to the domain■. Then, by applying the concentration-compactness principle for fractional Sobolev spaces in unbounded domains, we prove the existence res... 相似文献
2.
The purpose of this paper is threefold. First, we prove sharp singular affine Moser–Trudinger inequalities on both bounded and unbounded domains in \({\mathbb {R}}^{n}\). In particular, we will prove the following much sharper affine Moser–Trudinger inequality in the spirit of Lions (Rev Mat Iberoamericana 1(2):45–121, 1985) (see our Theorem 1.4): Let \(\alpha _{n}=n\left( \frac{n\pi ^{\frac{n}{2}}}{\Gamma (\frac{n}{2}+1)}\right) ^{\frac{1}{n-1}}\), \(0\le \beta <n\) and \(\tau >0\). Then there exists a constant \(C=C\left( n,\beta \right) >0\) such that for all \(0\le \alpha \le \left( 1-\frac{\beta }{n}\right) \alpha _{n}\) and \(u\in C_{0}^{\infty }\left( {\mathbb {R}}^{n}\right) \setminus \left\{ 0\right\} \) with the affine energy \(~{\mathcal {E}}_{n}\left( u\right) <1\), we have Moreover, the constant \(\left( 1-\frac{\beta }{n}\right) \alpha _{n}\) is the best possible in the sense that there is no uniform constant \(C(n, \beta )\) independent of u in the above inequality when \(\alpha >\left( 1-\frac{\beta }{n}\right) \alpha _{n}\). Second, we establish the following improved Adams type inequality in the spirit of Lions (Theorem 1.8): Let \(0\le \beta <2m\) and \(\tau >0\). Then there exists a constant \(C=C\left( m,\beta ,\tau \right) >0\) such that for all \(0\le \alpha \le \left( 1-\frac{\beta }{2m}\right) \beta (2m,2)\). When \(\alpha >\left( 1-\frac{\beta }{2m}\right) \beta (2m,2)\), the supremum is infinite. In the above, we use The main difficulties of proving the above results are that the symmetrization method does not work. Therefore, our main ideas are to develop a rearrangement-free argument in the spirit of Lam and Lu (J Differ Equ 255(3):298–325, 2013; Adv Math 231(6): 3259–3287, 2012), Lam et al. (Nonlinear Anal 95: 77–92, 2014) to establish such theorems. Third, as an application, we will study the existence of weak solutions to the biharmonic equation where the nonlinearity f has the critical exponential growth.
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$$\begin{aligned} {\displaystyle \int \nolimits _{{\mathbb {R}}^{n}}} \frac{\phi _{n,1}\left( \frac{2^{\frac{1}{n-1}}\alpha }{\left( 1+{\mathcal {E}}_{n}\left( u\right) ^{n}\right) ^{\frac{1}{n-1}}}\left| u\right| ^{\frac{n}{n-1}}\right) }{\left| x\right| ^{\beta }}dx\le C\left( n,\beta \right) \frac{\left\| u\right\| _{n}^{n-\beta }}{\left| 1-{\mathcal {E}}_{n}\left( u\right) ^{n}\right| ^{1-\frac{\beta }{n}}}. \end{aligned}$$
$$\begin{aligned} \underset{u\in W^{2,m}\left( {\mathbb {R}}^{2m}\right) , \int _{ {\mathbb {R}}^{2m}}\left| \Delta u\right| ^{m}+\tau \left| u\right| ^{m} \le 1}{\sup } {\displaystyle \int \nolimits _{{\mathbb {R}}^{2m}}} \frac{\phi _{2m,2}\left( \frac{2^{\frac{1}{m-1}}\alpha }{\left( 1+\left\| \Delta u\right\| _{m}^{m}\right) ^{\frac{1}{m-1}}}\left| u\right| ^{\frac{m}{m-1}}\right) }{\left| x\right| ^{\beta }}dx\le C\left( m,\beta ,\tau \right) , \end{aligned}$$
$$\begin{aligned} \phi _{p,q}(t)=e^{t}- {\displaystyle \sum \limits _{j=0}^{j_{\frac{p}{q}}-2}} \frac{t^{j}}{j!},\,\,\,j_{\frac{p}{q}}=\min \left\{ j\in {\mathbb {N}} :j\ge \frac{p}{q}\right\} \ge \frac{p}{q}. \end{aligned}$$
$$\begin{aligned} \left\{ \begin{array}{l} \Delta ^{2}u+V(x)u=f(x,u)\text { in }{\mathbb {R}}^{4}\\ u\in H^{2}\left( {\mathbb {R}}^{4}\right) ,~u\ge 0 \end{array} \right. , \end{aligned}$$
3.
Oleksandr Misiats Oleksandr Stanzhytskyi Nung Kwan Yip 《Journal of Theoretical Probability》2016,29(3):996-1026
In this paper, we investigate the long-time behavior of stochastic reaction–diffusion equations of the type \(\text {d}u = (Au + f(u))\text {d}t + \sigma (u) \text {d}W(t)\), where \(A\) is an elliptic operator, \(f\) and \(\sigma \) are nonlinear maps and \(W\) is an infinite-dimensional nuclear Wiener process. The emphasis is on unbounded domains. Under the assumption that the nonlinear function \(f\) possesses certain dissipative properties, this equation is known to have a solution with an expectation value which is uniformly bounded in time. Together with some compactness property, the existence of such a solution implies the existence of an invariant measure, which is an important step in establishing the ergodic behavior of the underlying physical system. In this paper, we expand the existing classes of nonlinear functions \(f\) and \(\sigma \) and elliptic operators \(A\) for which the invariant measure exists, in particular in unbounded domains. We also show the uniqueness of the invariant measure for an equation defined on the upper half space if \(A\) is the Shrödinger-type operator \(A = \frac{1}{\rho }(\text {div} \rho \nabla u)\) where \(\rho = \text {e}^{-|x|^2}\) is the Gaussian weight. 相似文献
4.
The finite-dimensional variational inequality problem (VIP) has been studied extensively in the literature because of its successful applications in many fields such as economics, transportation, regional science and operations research. Barker and Pang[1] have given an excellent survey of theories, methods and applications of VIPs. 相似文献
5.
Generalizations of the Trudinger–Moser inequality to Sobolev spaces with singular weights are considered for any smooth domain Ω???? N . Furthermore, we show that the resulting inequalities are sharp obtaining the best exponents. 相似文献
6.
In this paper,applying the concept of generalized KKM map,we study problems ofvariational inequalities.We weaken convexity(concavity)conditions for a functional of two variables■(x,y)in the general variational inequalities.Last,we show a proof of non-topological degree meth-od of acute angle principle about monotone operator as an application of these results. 相似文献
7.
We prove some uniform and pointwise gradient estimates for the Dirichlet and the Neumann evolution operators \(G_{\mathcal {D}}(t,s)\) and \(G_{\mathcal {N}}(t,s)\) associated with a class of nonautonomous elliptic operators (t) with unbounded coefficients defined in I× \(\mathbb{R}_{+}\) (where I is a right-halfline or I=?). We also prove the existence and the uniqueness of a tight evolution system of measures \(\left \{\mu _{t}^{\mathcal {N}}\right \}_{t \in I}\) associated with \(G_{\mathcal {N}}(t,s)\) , which turns out to be sub-invariant for \(G_{\mathcal {D}}(t,s)\) , and we study the asymptotic behaviour of the evolution operators \(G_{\mathcal {D}}(t,s)\) and \(G_{\mathcal {N}}(t,s)\) in the L p -spaces related to the system \(\left \{\mu _{t}^{\mathcal {N}}\right \}_{t \in I}\) . 相似文献
8.
Thispaperiscontinuationsof[1],westilltousemarksandtermsin[1]andtheothertermsagreewith[2]and[3].LetEandFbetwoHausdorfftopologicalvectorspaces,XE,YFbetwononemptysets,FbethedualspaceofF,A:X→2YandB:Y→2Fbetwoset-valuedmapping,T:Y→Xbeinvertible.Inthisp… 相似文献
9.
Boundedness of Solutions for Elliptic Variational InequalitiesBoundednessofSolutionsforEllipticVariationalInequalities¥YeRuif... 相似文献
10.
We consider the problem of the existence of bi-cores for some classes of second order elliptic differential operators with unbounded coefficients generating bi-continuous semigroups on the space of bounded continuous functions on N. 相似文献
11.
V. I. Ivanov 《Journal of Optimization Theory and Applications》2010,146(3):602-616
In the present paper we consider a pseudoconvex (in an extended sense) function f using higher order Dini directional derivatives. A Variational Inequality, which is a refinement of the Stampacchia Variational
Inequality, is defined. We prove that the solution set of this problem coincides with the set of global minimizers of f if and only if f is pseudoconvex. We introduce a notion of pseudomonotone Dini directional derivatives (in an extended sense). It is applied
to prove that the solution sets of the Stampacchia Variational Inequality and Minty Variational Inequality coincide if and
only if the function is pseudoconvex. At last, we obtain several characterizations of the solution set of a program with a
pseudoconvex objective function. 相似文献
12.
We give a sufficient condition for a class of jump-type symmetric Dirichlet forms on ? d to be conservative in terms of the jump kernel and the associated measure. Our condition allows the coefficients dominating big jumps to be unbounded. We derive the conservativeness for Dirichlet forms related to symmetric stable processes. We also show that our criterion is sharp by using time changed Dirichlet forms. We finally remark that our approach is applicable to jump-diffusion type symmetric Dirichlet forms on ? d . 相似文献
13.
This paper establishes a bridge between set optimization problems and vector Ky Fan inequality problems. We introduce a general model, called the bifunction-set optimization problem, that provides a unifying framework for the above-mentioned problems. An existence result in our model is obtained, with the help of KKM–Fan’s lemma. As applications, we derive some new or sharper existence results for set optimization problems and generalized vector Ky Fan inequalities with efficient solutions. 相似文献
14.
V. V. Kozlov 《Siberian Mathematical Journal》2018,59(6):1090-1093
We discuss the problem of existence of a smooth endomorphism of a closed n-dimensional manifold carrying a differential n-form into a prescribed volume form. Of course, we assume that the integrals of these forms over the whole manifold are equal. The solution of this problem for the n-dimensional torus reduces to the problem of small denominators well known in analysis. 相似文献
15.
Kyeong-Hun Kim 《Journal of Theoretical Probability》2009,22(1):220-238
Stochastic partial differential equations of divergence form are considered on Lipschitz domains. Existence and uniqueness
results are given in weighted Sobolev spaces. It is allowed that the coefficients of the equations substantially oscillate
or blow up near the boundary.
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16.
The existence and uniqueness of the global solution of stochastic differential equations with discrete variable delay is investigated
in this paper, and the pathwise estimation is also done by using Lyapunov function method and exponential martingale inequality.
The results can be used not only in the case of bounded delay but also in the case of unbounded delay. As the applications,
this paper considers the pathwise estimation of solutions of stochastic pantograph equations. 相似文献
17.
Shao Lin JI Zhen WU 《数学学报(英文版)》2007,23(12):2189-2204
The authors get a maximum principle for one kind of stochastic optimization problem motivated by dynamic measure of risk. The dynamic measure of risk to an investor in a financial market can be studied in our framework where the wealth equation may have nonlinear coefficients. 相似文献
18.
Convergence of Hybrid Steepest-Descent Methods for Generalized Variational Inequalities 总被引:3,自引:0,他引:3
Liu Chuan ZENG N. C. Wong J. C. YAO 《数学学报(英文版)》2006,22(1):1-12
In this paper, we consider the generalized variational inequality GVI(F, g, C), where F and g are mappings from a Hilbert space into itself and C is the fixed point set of a nonexpansive mapping. We propose two iterative algorithms to find approximate solutions of the GVI(F,g, C). Strong convergence results are established and applications to constrained generalized pseudo-inverse are included. 相似文献
19.
We consider the following initial boundary problem of derivative complex Ginzburg-Landau (DCGL) equation 相似文献
20.
Bo Ling 《Numerical Functional Analysis & Optimization》2013,34(6):724-738
We consider the optimal recovery problem of isotropic classes of r-th differentiable multivariate functions defined on ? d , and obtain some asymptotically optimal results. It turns out that this optimal recovery problem is intimately related to the optimal covering problem of ? d by equal balls in discrete geometry. 相似文献