Infinitely many low- and high-energy solutions for a class of elliptic equations with variable exponent

This paper is concerned with the $p(x)$-Laplacian equation of the form $$ \left\{\begin{array}{ll} -\Delta_{p(x)} u=Q(x)|u|^{r(x)-2}u, &\mbox{in}\ \Omega,\u=0, &\mbox{on}\ \partial \Omega, \end{array}\right. \eqno{0.1} $$ where $\Omega\subset\R^N$ is a smooth bounded domain, $1p^+$ and $Q: \overline{\Omega}\to\R$ is a nonnegative continuous function. We prove that (0.1) has infinitely many small solutions and infinitely many large solutions by using the Clark''s theorem and the symmetric mountain pass lemma.     

Multiplicity of Weak Solutions for a $(p(x), q(x))$-Kirchhoff Equation with Neumann Boundary Conditions

The aim of this study is to investigate the existence of infinitely many weak solutions for the $(p(x), q(x))$-Kirchhoff Neumann problem described by the following equation : \begin{equation*} \left\{\begin{array}{ll} -\left(a_{1}+a_{2}\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx\right)\Delta_{p(\cdot)}u-\left(b_{1}+b_{2}\int_{\Omega}\frac{1}{q(x)}|\nabla u|^{q(x)}dx\right)\Delta_{q(\cdot)}u\+\lambda(x)\Big(|u|^{p(x)-2} u+|u|^{q(x)-2} u\Big)= f_1(x,u)+f_2(x,u) &\mbox{ in } \Omega, \\frac{\partial u}{\partial \nu} =0 \quad &\mbox{on} \quad \partial\Omega.\end{array}\right. \end{equation*} By employing a critical point theorem proposed by B. Ricceri, which stems from a more comprehensive variational principle, we have successfully established the existence of infinitely many weak solutions for the aforementioned problem.     

Quadratic derivative nonlinear Schrödinger equations in two space dimensions

We study the global in time existence of small classical solutions to the nonlinear Schrödinger equation with quadratic interactions of derivative type in two space dimensions $\left\{\begin{array}{l@{\quad}l}i \partial _{t} u+\frac{1}{2}\Delta u=\mathcal{N}\left( \nabla u,\nabla u\right),&;t >0 ,\;x\in {\bf R}^{2},\\ u\left( 0,x\right) =u_{0} \left( x\right),&;x\in {\bf R}^{2}, \end{array}\right.\quad\quad\quad\quad\quad\quad (0.1)$ where the quadratic nonlinearity has the form ${\mathcal{N}( \nabla u,\nabla v) =\sum_{k,l=1,2}\lambda _{kl} (\partial _{k}u) ( \partial _{l}v) }We study the global in time existence of small classical solutions to the nonlinear Schr?dinger equation with quadratic interactions of derivative type in two space dimensions

$\left\{{l@{\quad}l}i \partial _{t} u+\frac{1}{2}\Delta u=\mathcal{N}\left( \nabla u,\nabla u\right),&t >0 ,\;x\in {\bf R}^{2},\\ u\left( 0,x\right) =u_{0} \left( x\right),&x\in {\bf R}^{2}, \right.\quad\quad\quad\quad\quad\quad (0.1)$\left\{\begin{array}{l@{\quad}l}i \partial _{t} u+\frac{1}{2}\Delta u=\mathcal{N}\left( \nabla u,\nabla u\right),&t >0 ,\;x\in {\bf R}^{2},\\ u\left( 0,x\right) =u_{0} \left( x\right),&x\in {\bf R}^{2}, \end{array}\right.\quad\quad\quad\quad\quad\quad (0.1)      4. Existence and concentration result for Kirchhoff equations with critical exponent and Hartree nonlinearity          下载免费PDF全文 Guofeng Che  Haibo Chen 《Journal of Applied Analysis & Computation》2020,10(5):2121-2144 This paper is concerned with the following Kirchhoff-type equations $$ \left\{ \begin{array}{ll} \displaystyle -\big(\varepsilon^{2}a+\varepsilon b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\mathrm{d}x\big)\Delta u + V(x)u+\mu\phi |u|^{p-2}u=f(x,u), &\quad \mbox{ in }\mathbb{R}^{3},\(-\Delta)^{\frac{\alpha}{2}} \phi=\mu|u|^{p},~u>0, &\quad \mbox{ in }\mathbb{R}^{3},\\end{array} \right. $$ where $f(x,u)=\lambda K(x)|u|^{q-2}u+Q(x)|u|^{4}u$, $a>0,~b,~\mu\geq0$ are constants, $\alpha\in(0,3)$, $p\in[2,3),~q\in[2p,6)$ and $\varepsilon,~\lambda>0$ are parameters. Under some mild conditions on $V(x),~K(x)$ and $Q(x)$, we prove that the above system possesses a ground state solution $u_{\varepsilon}$ with exponential decay at infinity for $\lambda>0$ and $\varepsilon$ small enough. Furthermore, $u_{\varepsilon}$ concentrates around a global minimum point of $V(x)$ as $\varepsilon\rightarrow0$. The methods used here are based on minimax theorems and the concentration-compactness principle of Lions. Our results generalize and improve those in Liu and Guo (Z Angew Math Phys 66: 747-769, 2015), Zhao and Zhao (Nonlinear Anal 70: 2150-2164, 2009) and some other related literature.      5. 含有Sobolev-Hardy临界指标的奇异椭圆方程Neumann问题无穷多解的存在性          下载免费PDF全文 胡爱莲  张正杰 《数学物理学报(A辑)》2007,27(6):1025-1034 该文研究了如下的奇异椭圆方程Neumann问题$\left\{\begin{array}{ll}\disp -\Delta u-\frac{\mu u}{|x|^2}=\frac{|u|^{2^{*}(s)-2}u}{|x|^s}+\lambda|u|^{q-2}u,\ \ &;x\in\Omega,\\D_\gamma{u}+\alpha(x)u=0,&;x\in\partial\Omega\backslash\{0\},\end{array}\right.$其中$\Omega $ 是 $ R^N$ 中具有 $ C^1$边界的有界区域, $ 0\in\partial\Omega$, $N\ge5$. $2^{*}(s)=\frac{2(N-s)}{N-2}$ (该文研究了如下的奇异椭圆方程Neumann问题$\left\{\begin{array}{ll}\disp -\Delta u-\frac{\mu u}{|x|^2}=\frac{|u|^{2^{*}(s)-2}u}{|x|^s}+\lambda|u|^{q-2}u,\ \ &;x\in\Omega,\\D_\gamma{u}+\alpha(x)u=0,&;x\in\partial\Omega\backslash\{0\},\end{array}\right.$其中$\Omega $ 是 $ R^N$ 中具有 $ C^1$边界的有界区域, $ 0\in\partial\Omega$, $N\ge5$. $2^{*}(s)=\frac{2(N-s)}{N-2}$ (该文研究了如下的奇异椭圆方程Neumann问题其中Ω是RN中具有C1边界的有界区域,0∈■Ω,N≥5.2*(s)=2(N-s)/N-2(0≤s≤2)是临界Sobolev-Hardy指标, 10.利用变分方法和对偶喷泉定理,证明了这个方程无穷多解的存在性.      6. 一类具有奇异灵敏度的logistic趋化系统解的渐近行为      杜宛娟 《数学研究及应用》2021,41(5):473-480 本文在无边界流的光滑有界区域$\Omega\subset\mathbb{R}^n~(n>2)$上研究了具有奇异灵敏度及logistic源的抛物-椭圆趋化系统$$\left\{\begin{array}{ll}u_t=\Delta u-\chi\nabla\cdot(\frac{u}{v}\nabla v)+r u-\mu u^k,&x\in\Omega,\,t>0,\\ 0=\Delta v-v+u,&x\in\Omega,\,t>0\end{array}\right.$$ 其中$\chi$, $r$, $\mu>0$, $k\geq2$. 证明了若当$r$适当大, 则当$t\rightarrow\infty$时该趋化系统全局有界解呈指数收敛于$((\frac{r}{\mu})^{\frac{1}{k-1}}, (\frac{r}{\mu})^{\frac{1}{k-1}})$.      7. 具有凹凸非线性项和变号位势函数拟线性椭圆系统解的多重结果      储昌木  唐春雷 《数学年刊A辑(中文版)》2011,32(4):443-458 研究拟线性椭圆系统(?)的非平凡非负解或正解的多重性,这里Ω(?)R~N是具有光滑边界(?)Ω的有界域,1≤qp~*/p~*-q,其中当N≤p时,p~*=+∞,而当1相似文献    8. Error function inequalities      Horst Alzer 《Advances in Computational Mathematics》2010,33(3):349-379 We present various inequalities for the error function. One of our theorems states: Let α?≥?1. For all x,y?>?0 we have $$ \delta_{\alpha} < \frac{ \mbox{erf} \left( x+ \mbox{erf}(y)^{\alpha}\right) +\mbox{erf}\left( y+ \mbox{erf}(x)^{\alpha}\right) } {\mbox{erf}\left( \mbox{erf}(x)+\mbox{erf}(y)\right) } < \Delta_{\alpha} $$ with the best possible bounds $$ \delta_{\alpha}= \left\{ \begin{array}{ll} 1+\sqrt{\pi}/2, & \ \ \textrm{{if} $\alpha=1$,}\\ \sqrt{\pi}/2, & \ \ \textrm{{if} $\alpha>1$,}\\ \end{array}\right. \quad{\mbox{and} \,\,\,\,\, \Delta_{\alpha}=1+\frac{1}{\mbox{erf}(1)}.} $$       9. Existence of solutions of the Neumann problem for a class of equations involving the p-Laplacian via a variational principle of Ricceri      G. Anello  G. Cordaro 《Archiv der Mathematik》2002,79(4):274-287 In this paper we deal with the existence of weak solutions for the following Neumann problem¶¶$ \left\{{ll} -\mathrm{div}(|\nabla u|^{p-2}\nabla u) + \lambda(x)|u|^{p-2}u = \alpha(x)f(u) + \beta(x)g(u) $ \left\{\begin{array}{ll} -\mathrm{div}(|\nabla u|^{p-2}\nabla u) + \lambda(x)|u|^{p-2}u = \alpha(x)f(u) + \beta(x)g(u) &; $ \mbox{in $ \mbox{in \Omega$}\\ {\partial u \over \partial \nu} = 0 $}\\ {\partial u \over \partial \nu} = 0 &; $ \mbox{on $ \mbox{on \partial \Omega$} \right. $}\end{array} \right. ¶¶ where $ \nu $ \nu is the outward unit normal to the boundary $ \partial\Omega $ \partial\Omega of the bounded open set _boxclose^N \Omega \subset \mathbb{R}^N . The existence of solutions, for the above problem, is proved by applying a critical point theorem recently obtained by B. Ricceri as a consequence of a more general variational principle.      10. Infinitely many solutions for a zero mass Schodinger-Poisson-Slater problem with critical growth          下载免费PDF全文 Liu Yang  Zhisu Liu 《Journal of Applied Analysis & Computation》2019,9(5):1706-1718 In this paper, we are concerned with the following Schr\"{o}dinger-Poisson-Slater problem with critical growth: $$ -\Delta u+(u^{2}\star \frac{1}{|4\pi x|})u=\mu k(x)|u|^{p-2}u+|u|^{4}u\,\,\mbox{in}\,\,\R^{3}. $$ We use a measure representation concentration-compactness principle of Lions to prove that the $(PS)_{c}$ condition holds locally. Via a truncation technique and Krasnoselskii genus theory, we further obtain infinitely many solutions for $\mu\in(0,\mu^{\ast})$ with some $\mu^{\ast}>0$.      11. 一类Kirchhoff型系统解的存在性      刘晓敏  杨作东 《数学研究及应用》2018,38(4):411-417 In this paper,we are interested in the existence of positive solutions for the Kirchhoff type problems{-(a_1 + b_1M_1(∫_?|▽u|~pdx))△_(_pu) = λf(u,v),in ?,-(a_2 + b_2M_2(∫?|▽v|~qdx))△_(_qv) = λg(u,v),in ?,u = v = 0,on ??,where 1 p,q N,M i:R_0~+→ R~+(i = 1,2) are continuous and increasing functions.λ is a parameter,f,g ∈ C~1((0,∞) ×(0,∞)) × C([0,∞) × [0,∞)) are monotone functions such that f_s,f_t,g_s,g_t ≥ 0,and f(0,0) 0,g(0,0) 0(semipositone).Our proof is based on the sub-and super-solutions techniques.      12. 重调和方程非平凡解的存在性          下载免费PDF全文 唐春霞  张正杰 《数学物理学报(A辑)》2008,28(2):256-265 该文主要研究$R^N(N>4)$上重调和方程\begin{eqnarray*}\left\{\begin{array}{ll} \Delta^2 u+\lambda u=\overline{f}(x,u);\\ \lim\limits_{|x|\rightarrow\infty}u(x)=0;\\u\in{H^2}(R^N),\hspace{0.1cm}x\in{R^N } \end{array}\right.\end{eqnarray*}的非平凡解的存在性.为了便于研究,将方程转化为$R^N(N>4)$ 上带有扰动项的重调和方程\begin{eqnarray*}\left\{\begin{array}{ll} \Delta^2 u+\lambda u=f(u)+\varepsilon g(x,u);\\ \lim\limits_{|x|\rightarrow\infty}u(x)=0;\\u\in{H^2}(R^N),\hspace{0.1cm}x\in{R^N } .\end{array}\right.\end{eqnarray*}并运用扰动方法进行研究(其中$f(u)=\lim\limits_{|x|\longrightarrow \infty}\overline{f}(x,u),\varepsilon g(x,u)=\overline{f}(x,u)-f(u),\varepsilon$为任意小常数),证明了在适当条件下上述问题非平凡解的存在性.      13. Gelfand type problem for singular quadratic quasilinear equations      Alexis Molino 《NoDEA : Nonlinear Differential Equations and Applications》2016,23(5):56 In this paper, we study the existence of positive solutions for the quasilinear elliptic singular problem $$\left\{\begin{array}{ll}-\Delta u + c\,\frac{|\nabla u|^2}{u^\gamma} = \lambda\,f(u), \quad \quad \mbox{in $\Omega$},\\ u=0, \quad \qquad \qquad \qquad \quad \, \, \, \, \, \mbox{on $\partial$$\Omega$},\end{array}\right.$$ where \({c,\lambda >0}\), \({\gamma \in (0,1)}\), f is strictly increasing and derivable in \({[0,\infty)}\) with \({f(0)>0}\). We show that there exists \({\lambda^*>0}\) such that \({(0,\lambda^*]}\) is the maximal set of values such there exists solution. In addition, we prove that for \({\lambda<\lambda^*}\) there exists minimal and bounded solutions. Moreover, we give sufficient conditions for existence and regularity of solutions for \({\lambda=\lambda^*}\).       14. Entropy Numbers of Embeddings of Weighted Besov Spaces      Thomas Kuhn  Hans-Gerd Leopold  Winfried Sickel  Leszek Skrzypczak 《Constructive Approximation》2005,23(1):61-77 We investigate the asymptotic behavior of the entropy numbers of the compact embedding $$ B^{s_1}_{p_1,q_1} \!\!(\mbox{\footnotesize\bf R}^d, \alpha) \hookrightarrow B^{s_2}_{p_2,q_2} \!\!({\xxR}). $$ Here $B^s_{p,q} \!({\mbox{\footnotesize\bf R}^d}, \alpha)$ denotes a weighted Besov space, where the weight is given by $w_\alpha (x) = (1+| x |^2)^{\alpha/2}$, and $B^{s_2}_{p_2,q_2} \!({\mbox{\footnotesize\bf R}^d})$ denotes the unweighted Besov space, respectively. We shall concentrate on the so-called limiting situation given by the following constellation of parameters: $s_2 < s_1$, $0 < p_1,p_2 \le \infty$, and $$ \alpha = s_1 - \frac{d}{p_1} - s_2 + \frac{d}{p_2} > d \, \max \Big(0, \frac{1}{p_2}-\frac{1}{p_1}\Big). $$ In almost all cases we give a sharp two-sided estimate.      15. 三维液晶关于压力水平梯度的相关正则性准则      赵玲玲  李风泉 《数学研究及应用》2020,40(2):165-168 This paper is devoted to investigating regularity criteria for the 3-D nematic liquid crystal flows in terms of horizontal derivative components of the pressure and gradient of the orientation field. More precisely, we mainly proved that the strong solution(u, d)can be extended beyond T, provided that the horizontal derivative components of the pressure■ and gradient of the orientation field satisfy■ and■.      16. Uniqueness and existence of solutions for a singular system with nonlocal operator via perturbation method          下载免费PDF全文 Kamel Saoudi  Mouna Kratou  Eadah AlZahrani 《Journal of Applied Analysis & Computation》2020,10(4):1311-1325 In this work, we investigate the existence and the uniqueness of solutions for the nonlocal elliptic system involving a singular nonlinearity as follows: $$ \left\{\begin{array}{ll} (-\Delta_p)^su = a(x)|u|^{q-2}u +\frac{1-\alpha}{2-\alpha-\beta} c(x)|u|^{-\alpha}|v|^{1-\beta}, \quad \text{in }\Omega,\ (-\Delta_p)^s v= b(x)|v|^{q-2}v +\frac{1-\beta}{2-\alpha-\beta} c(x)|u|^{1-\alpha}|v|^{-\beta}, \quad \text{in }\Omega,\ u=v = 0 ,\;\;\mbox{ in }\,\mathbb{R}^N\setminus\Omega, \end{array} \right. $$ where $\Omega $ is a bounded domain in $\mathbb{R}^{n}$ with smooth boundary, $0<\alpha <1,$ $0<\beta <1,$ $2-\alpha -\beta 相似文献    17. A NECESSARY AND SUFFICIENT CONDITION FOR THE OSCILLATION OF HIGHER-ORDER NEUTRAL EQUATIONS WITH SEVERAL DELAYS      Wang Zhicheng 《数学年刊B辑(英文版)》1991,12(3):243-254 Consider the higher-order neutral delay differential equationd~t/dt~n(x(t)+sum from i=1 to lp_ix(t-τ_i)-sum from j=1 to mr_jx(t-ρ_j))+sum from k=1 to Nq_kx(t-u_k)=0,(A)where the coefficients and the delays are nonnegative constants with n≥2 even. Then anecessary and sufficient condition for the oscillation of (A) is that the characteristicequationλ~n+λ~nsum from i=1 to lp_ie~(-λτ_i-λ~n)sum from j=1 to mr_je~(-λρ_j)+sum from k=1 to Nq_ke~(-λρ_k)=0has no real roots.      18. THE ASYMPTOTIC BEHAVIOUR OF ANALYTIC FUNCTIONS      Hu Ke 《数学年刊B辑(英文版)》1983,4(2):187-190 AIn this paper, the author obtains the following results:(1) If Taylor coeffiients of a function satisfy the conditions:(i),(ii),(iii)A_k=O(1/k) the for any h>0 the function φ(z)=exp{w(z)} satisfies the asymptotic equality the case h>1/2 was proved by Milin.(2) If f(z)=z α_2z~2 …∈S~* and,then for λ>1/2      19. Supercritical Elliptic Equation in Hyperbolic Space          下载免费PDF全文 Haiyang He 《偏微分方程(英文版)》2015,28(2):120-127 In this paper, we study the following semi-linear elliptic equation $$-Δ_H^nu=|u|^{p-2}u,\qquad\qquad (0.1)$$ in the whole Hyperbolic space $\mathbb{H}^n$,where n ≥ 3, p › 2n/(n-2). We obtain some regularity results for the radial singular solutions of problem (0.1). We show that the singular solution $u^∗$ with $lim_{t → 0}(sinht)^{\frac{2}{p-2}}⋅u(t)=±(\frac{2}{p-2}(n-2-\frac{2}{p-2})^{\frac{1}{p-2}}$ belongs to the closure (in the natural topology given by $H¹_{loc}(\mathbb{H}^N)∩L^p_{loc}(H^N))$ of the set of smooth classical solutions to the Eq. (0.1). In contrast, we also prove that any oscillating radial solutions of (0.1) on $\mathbb{H}^N$\{0} fails to be in the space $H¹_{loc}(\mathbb{H}^N)∩L^p_{loc}(H^N)$.      20. Inequalities and Separation for a Biharmonic Laplace-Beltrami Differential Operator in a Hilbert Space Associated with the Existence and Uniqueness Theorem          下载免费PDF全文 Elsayed M. E. Zayed 《偏微分方程(英文版)》2016,29(1):59-70 In this paper, we have studied the separation for the biharmonic Laplace-Beltrami differential operator\begin{equation*}Au(x)=-\Delta \Delta u(x)+V(x)u(x),\end{equation*}for all $x\in R^{n}$, in the Hilbert space $H=L_{2}(R^{n},H_{1})$ with the operator potential $V(x)\in C^{1}(R^{n},L(H_{1}))$, where $L(H_{1})$ is the space of all bounded linear operators on the Hilbert space $H_{1}$, while $\Delta \Delta u$\ is the biharmonic differential operator and\begin{equation*}\Delta u{=-}\sum_{i,j=1}^{n}\frac{1}{\sqrt{\det g}}\frac{\partial }{{\partial x_{i}}}\left[ \sqrt{\det g}g^{-1}(x)\frac{\partial u}{{\partial x}_{j}}\right]\end{equation*}is the Laplace-Beltrami differential operator in $R^{n}$. Here $g(x)=(g_{ij}(x))$ is the Riemannian matrix, while $g^{-1}(x)$ is the inverse of the matrix $g(x)$. Moreover, we have studied the existence and uniqueness Theorem for the solution of the non-homogeneous biharmonic Laplace-Beltrami differential equation $Au=-\Delta \Delta u+V(x)u(x)=f(x)$ in the Hilbert space $H$ where $f(x)\in H$ as an application of the separation approach.

Existence and concentration result for Kirchhoff equations with critical exponent and Hartree nonlinearity

This paper is concerned with the following Kirchhoff-type equations $$ \left\{ \begin{array}{ll} \displaystyle -\big(\varepsilon^{2}a+\varepsilon b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\mathrm{d}x\big)\Delta u + V(x)u+\mu\phi |u|^{p-2}u=f(x,u), &\quad \mbox{ in }\mathbb{R}^{3},\(-\Delta)^{\frac{\alpha}{2}} \phi=\mu|u|^{p},~u>0, &\quad \mbox{ in }\mathbb{R}^{3},\\end{array} \right. $$ where $f(x,u)=\lambda K(x)|u|^{q-2}u+Q(x)|u|^{4}u$, $a>0,~b,~\mu\geq0$ are constants, $\alpha\in(0,3)$, $p\in[2,3),~q\in[2p,6)$ and $\varepsilon,~\lambda>0$ are parameters. Under some mild conditions on $V(x),~K(x)$ and $Q(x)$, we prove that the above system possesses a ground state solution $u_{\varepsilon}$ with exponential decay at infinity for $\lambda>0$ and $\varepsilon$ small enough. Furthermore, $u_{\varepsilon}$ concentrates around a global minimum point of $V(x)$ as $\varepsilon\rightarrow0$. The methods used here are based on minimax theorems and the concentration-compactness principle of Lions. Our results generalize and improve those in Liu and Guo (Z Angew Math Phys 66: 747-769, 2015), Zhao and Zhao (Nonlinear Anal 70: 2150-2164, 2009) and some other related literature.     

含有Sobolev-Hardy临界指标的奇异椭圆方程Neumann问题无穷多解的存在性

该文研究了如下的奇异椭圆方程Neumann问题$\left\{\begin{array}{ll}\disp -\Delta u-\frac{\mu u}{|x|^2}=\frac{|u|^{2^{*}(s)-2}u}{|x|^s}+\lambda|u|^{q-2}u,\ \ &;x\in\Omega,\\D_\gamma{u}+\alpha(x)u=0,&;x\in\partial\Omega\backslash\{0\},\end{array}\right.$其中$\Omega $ 是 $ R^N$ 中具有 $ C^1$边界的有界区域, $ 0\in\partial\Omega$, $N\ge5$. $2^{*}(s)=\frac{2(N-s)}{N-2}$ (该文研究了如下的奇异椭圆方程Neumann问题$\left\{\begin{array}{ll}\disp -\Delta u-\frac{\mu u}{|x|^2}=\frac{|u|^{2^{*}(s)-2}u}{|x|^s}+\lambda|u|^{q-2}u,\ \ &;x\in\Omega,\\D_\gamma{u}+\alpha(x)u=0,&;x\in\partial\Omega\backslash\{0\},\end{array}\right.$其中$\Omega $ 是 $ R^N$ 中具有 $ C^1$边界的有界区域, $ 0\in\partial\Omega$, $N\ge5$. $2^{*}(s)=\frac{2(N-s)}{N-2}$ (该文研究了如下的奇异椭圆方程Neumann问题其中Ω是RN中具有C1边界的有界区域,0∈■Ω,N≥5.2*(s)=2(N-s)/N-2(0≤s≤2)是临界Sobolev-Hardy指标, 10.利用变分方法和对偶喷泉定理,证明了这个方程无穷多解的存在性.     

一类具有奇异灵敏度的logistic趋化系统解的渐近行为

本文在无边界流的光滑有界区域$\Omega\subset\mathbb{R}^n~(n>2)$上研究了具有奇异灵敏度及logistic源的抛物-椭圆趋化系统$$\left\{\begin{array}{ll}u_t=\Delta u-\chi\nabla\cdot(\frac{u}{v}\nabla v)+r u-\mu u^k,&x\in\Omega,\,t>0,\\ 0=\Delta v-v+u,&x\in\Omega,\,t>0\end{array}\right.$$ 其中$\chi$, $r$, $\mu>0$, $k\geq2$. 证明了若当$r$适当大, 则当$t\rightarrow\infty$时该趋化系统全局有界解呈指数收敛于$((\frac{r}{\mu})^{\frac{1}{k-1}}, (\frac{r}{\mu})^{\frac{1}{k-1}})$.     

具有凹凸非线性项和变号位势函数拟线性椭圆系统解的多重结果

研究拟线性椭圆系统(?)的非平凡非负解或正解的多重性,这里Ω(?)R~N是具有光滑边界(?)Ω的有界域,1≤qp~*/p~*-q,其中当N≤p时,p~*=+∞,而当1相似文献   

Error function inequalities

We present various inequalities for the error function. One of our theorems states: Let α?≥?1. For all x,y?>?0 we have $$ \delta_{\alpha} < \frac{ \mbox{erf} \left( x+ \mbox{erf}(y)^{\alpha}\right) +\mbox{erf}\left( y+ \mbox{erf}(x)^{\alpha}\right) } {\mbox{erf}\left( \mbox{erf}(x)+\mbox{erf}(y)\right) } < \Delta_{\alpha} $$ with the best possible bounds $$ \delta_{\alpha}= \left\{ \begin{array}{ll} 1+\sqrt{\pi}/2, & \ \ \textrm{{if} $\alpha=1$,}\\ \sqrt{\pi}/2, & \ \ \textrm{{if} $\alpha>1$,}\\ \end{array}\right. \quad{\mbox{and} \,\,\,\,\, \Delta_{\alpha}=1+\frac{1}{\mbox{erf}(1)}.} $$      

Existence of solutions of the Neumann problem for a class of equations involving the p-Laplacian via a variational principle of Ricceri

In this paper we deal with the existence of weak solutions for the following Neumann problem¶¶$ \left\{{ll} -\mathrm{div}(|\nabla u|^{p-2}\nabla u) + \lambda(x)|u|^{p-2}u = \alpha(x)f(u) + \beta(x)g(u) $ \left\{\begin{array}{ll} -\mathrm{div}(|\nabla u|^{p-2}\nabla u) + \lambda(x)|u|^{p-2}u = \alpha(x)f(u) + \beta(x)g(u) &; $ \mbox{in $ \mbox{in \Omega$}\\ {\partial u \over \partial \nu} = 0 $}\\ {\partial u \over \partial \nu} = 0 &; $ \mbox{on $ \mbox{on \partial \Omega$} \right. $}\end{array} \right. ¶¶ where $ \nu $ \nu is the outward unit normal to the boundary $ \partial\Omega $ \partial\Omega of the bounded open set _boxclose^N \Omega \subset \mathbb{R}^N . The existence of solutions, for the above problem, is proved by applying a critical point theorem recently obtained by B. Ricceri as a consequence of a more general variational principle.     

Infinitely many solutions for a zero mass Schodinger-Poisson-Slater problem with critical growth

In this paper, we are concerned with the following Schr\"{o}dinger-Poisson-Slater problem with critical growth: $$ -\Delta u+(u^{2}\star \frac{1}{|4\pi x|})u=\mu k(x)|u|^{p-2}u+|u|^{4}u\,\,\mbox{in}\,\,\R^{3}. $$ We use a measure representation concentration-compactness principle of Lions to prove that the $(PS)_{c}$ condition holds locally. Via a truncation technique and Krasnoselskii genus theory, we further obtain infinitely many solutions for $\mu\in(0,\mu^{\ast})$ with some $\mu^{\ast}>0$.     

一类Kirchhoff型系统解的存在性

In this paper,we are interested in the existence of positive solutions for the Kirchhoff type problems{-(a_1 + b_1M_1(∫_?|▽u|~pdx))△_(_pu) = λf(u,v),in ?,-(a_2 + b_2M_2(∫?|▽v|~qdx))△_(_qv) = λg(u,v),in ?,u = v = 0,on ??,where 1 p,q N,M i:R_0~+→ R~+(i = 1,2) are continuous and increasing functions.λ is a parameter,f,g ∈ C~1((0,∞) ×(0,∞)) × C([0,∞) × [0,∞)) are monotone functions such that f_s,f_t,g_s,g_t ≥ 0,and f(0,0) 0,g(0,0) 0(semipositone).Our proof is based on the sub-and super-solutions techniques.     

重调和方程非平凡解的存在性

该文主要研究$R^N(N>4)$上重调和方程\begin{eqnarray*}\left\{\begin{array}{ll} \Delta^2 u+\lambda u=\overline{f}(x,u);\\ \lim\limits_{|x|\rightarrow\infty}u(x)=0;\\u\in{H^2}(R^N),\hspace{0.1cm}x\in{R^N } \end{array}\right.\end{eqnarray*}的非平凡解的存在性.为了便于研究,将方程转化为$R^N(N>4)$ 上带有扰动项的重调和方程\begin{eqnarray*}\left\{\begin{array}{ll} \Delta^2 u+\lambda u=f(u)+\varepsilon g(x,u);\\ \lim\limits_{|x|\rightarrow\infty}u(x)=0;\\u\in{H^2}(R^N),\hspace{0.1cm}x\in{R^N } .\end{array}\right.\end{eqnarray*}并运用扰动方法进行研究(其中$f(u)=\lim\limits_{|x|\longrightarrow \infty}\overline{f}(x,u),\varepsilon g(x,u)=\overline{f}(x,u)-f(u),\varepsilon$为任意小常数),证明了在适当条件下上述问题非平凡解的存在性.     

Gelfand type problem for singular quadratic quasilinear equations

In this paper, we study the existence of positive solutions for the quasilinear elliptic singular problem

$$\left\{\begin{array}{ll}-\Delta u + c\,\frac{|\nabla u|^2}{u^\gamma} = \lambda\,f(u), \quad \quad \mbox{in $\Omega$},\\ u=0, \quad \qquad \qquad \qquad \quad \, \, \, \, \, \mbox{on $\partial$$\Omega$},\end{array}\right.$$

where \({c,\lambda >0}\), \({\gamma \in (0,1)}\), f is strictly increasing and derivable in \({[0,\infty)}\) with \({f(0)>0}\). We show that there exists \({\lambda^*>0}\) such that \({(0,\lambda^*]}\) is the maximal set of values such there exists solution. In addition, we prove that for \({\lambda<\lambda^*}\) there exists minimal and bounded solutions. Moreover, we give sufficient conditions for existence and regularity of solutions for \({\lambda=\lambda^*}\).

     

Entropy Numbers of Embeddings of Weighted Besov Spaces

We investigate the asymptotic behavior of the entropy numbers of the compact embedding $$ B^{s_1}_{p_1,q_1} \!\!(\mbox{\footnotesize\bf R}^d, \alpha) \hookrightarrow B^{s_2}_{p_2,q_2} \!\!({\xxR}). $$ Here $B^s_{p,q} \!({\mbox{\footnotesize\bf R}^d}, \alpha)$ denotes a weighted Besov space, where the weight is given by $w_\alpha (x) = (1+| x |^2)^{\alpha/2}$, and $B^{s_2}_{p_2,q_2} \!({\mbox{\footnotesize\bf R}^d})$ denotes the unweighted Besov space, respectively. We shall concentrate on the so-called limiting situation given by the following constellation of parameters: $s_2 < s_1$, $0 < p_1,p_2 \le \infty$, and $$ \alpha = s_1 - \frac{d}{p_1} - s_2 + \frac{d}{p_2} > d \, \max \Big(0, \frac{1}{p_2}-\frac{1}{p_1}\Big). $$ In almost all cases we give a sharp two-sided estimate.     

三维液晶关于压力水平梯度的相关正则性准则

This paper is devoted to investigating regularity criteria for the 3-D nematic liquid crystal flows in terms of horizontal derivative components of the pressure and gradient of the orientation field. More precisely, we mainly proved that the strong solution(u, d)can be extended beyond T, provided that the horizontal derivative components of the pressure■ and gradient of the orientation field satisfy■ and■.     

Uniqueness and existence of solutions for a singular system with nonlocal operator via perturbation method

In this work, we investigate the existence and the uniqueness of solutions for the nonlocal elliptic system involving a singular nonlinearity as follows: $$ \left\{\begin{array}{ll} (-\Delta_p)^su = a(x)|u|^{q-2}u +\frac{1-\alpha}{2-\alpha-\beta} c(x)|u|^{-\alpha}|v|^{1-\beta}, \quad \text{in }\Omega,\ (-\Delta_p)^s v= b(x)|v|^{q-2}v +\frac{1-\beta}{2-\alpha-\beta} c(x)|u|^{1-\alpha}|v|^{-\beta}, \quad \text{in }\Omega,\ u=v = 0 ,\;\;\mbox{ in }\,\mathbb{R}^N\setminus\Omega, \end{array} \right. $$ where $\Omega $ is a bounded domain in $\mathbb{R}^{n}$ with smooth boundary, $0<\alpha <1,$ $0<\beta <1,$ $2-\alpha -\beta 相似文献   

A NECESSARY AND SUFFICIENT CONDITION FOR THE OSCILLATION OF HIGHER-ORDER NEUTRAL EQUATIONS WITH SEVERAL DELAYS

Consider the higher-order neutral delay differential equationd~t/dt~n(x(t)+sum from i=1 to lp_ix(t-τ_i)-sum from j=1 to mr_jx(t-ρ_j))+sum from k=1 to Nq_kx(t-u_k)=0,(A)where the coefficients and the delays are nonnegative constants with n≥2 even. Then anecessary and sufficient condition for the oscillation of (A) is that the characteristicequationλ~n+λ~nsum from i=1 to lp_ie~(-λτ_i-λ~n)sum from j=1 to mr_je~(-λρ_j)+sum from k=1 to Nq_ke~(-λρ_k)=0has no real roots.     

THE ASYMPTOTIC BEHAVIOUR OF ANALYTIC FUNCTIONS

AIn this paper, the author obtains the following results:(1) If Taylor coeffiients of a function satisfy the conditions:(i),(ii),(iii)A_k=O(1/k) the for any h>0 the function φ(z)=exp{w(z)} satisfies the asymptotic equality the case h>1/2 was proved by Milin.(2) If f(z)=z α_2z~2 …∈S~* and,then for λ>1/2     

Supercritical Elliptic Equation in Hyperbolic Space

In this paper, we study the following semi-linear elliptic equation $$-Δ_H^nu=|u|^{p-2}u,\qquad\qquad (0.1)$$ in the whole Hyperbolic space $\mathbb{H}^n$,where n ≥ 3, p › 2n/(n-2). We obtain some regularity results for the radial singular solutions of problem (0.1). We show that the singular solution $u^∗$ with $lim_{t → 0}(sinht)^{\frac{2}{p-2}}⋅u(t)=±(\frac{2}{p-2}(n-2-\frac{2}{p-2})^{\frac{1}{p-2}}$ belongs to the closure (in the natural topology given by $H¹_{loc}(\mathbb{H}^N)∩L^p_{loc}(H^N))$ of the set of smooth classical solutions to the Eq. (0.1). In contrast, we also prove that any oscillating radial solutions of (0.1) on $\mathbb{H}^N$\{0} fails to be in the space $H¹_{loc}(\mathbb{H}^N)∩L^p_{loc}(H^N)$.     

Inequalities and Separation for a Biharmonic Laplace-Beltrami Differential Operator in a Hilbert Space Associated with the Existence and Uniqueness Theorem

In this paper, we have studied the separation for the biharmonic Laplace-Beltrami differential operator\begin{equation*}Au(x)=-\Delta \Delta u(x)+V(x)u(x),\end{equation*}for all $x\in R^{n}$, in the Hilbert space $H=L_{2}(R^{n},H_{1})$ with the operator potential $V(x)\in C^{1}(R^{n},L(H_{1}))$, where $L(H_{1})$ is the space of all bounded linear operators on the Hilbert space $H_{1}$, while $\Delta \Delta u$\ is the biharmonic differential operator and\begin{equation*}\Delta u{=-}\sum_{i,j=1}^{n}\frac{1}{\sqrt{\det g}}\frac{\partial }{{\partial x_{i}}}\left[ \sqrt{\det g}g^{-1}(x)\frac{\partial u}{{\partial x}_{j}}\right]\end{equation*}is the Laplace-Beltrami differential operator in $R^{n}$. Here $g(x)=(g_{ij}(x))$ is the Riemannian matrix, while $g^{-1}(x)$ is the inverse of the matrix $g(x)$. Moreover, we have studied the existence and uniqueness Theorem for the solution of the non-homogeneous biharmonic Laplace-Beltrami differential equation $Au=-\Delta \Delta u+V(x)u(x)=f(x)$ in the Hilbert space $H$ where $f(x)\in H$ as an application of the separation approach.