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1.
We establish sufficient conditions under which the quasilinear equation $$-div(|∇u|^{n-2}∇u)+V(x)|u|^{n-2}u=\frac{f(x,u)}{|x|^β}+εh(x) in \mathbb{R}^n,$$ has at least two nontrivial weak solutions in $W^{1,n} (\mathbb{R}^n)$ when ε > 0 is small enough, 0≤β < n, V is a continuous potential, f(x,u) behaves like $exp{γ|u|^{n/(n-1)}}$ as $|u|→∞$ for some γ > 0 and h≢ 0 belongs to the dual space of $W^{1,n} (\mathbb{R}^n)$.  相似文献   

2.
In this paper we study the initial boundary value problem of GBBM equations on unbounded domain u_t - Δu_t = div f(u) u(x,0) = u_0(x) u|_{∂Ω} = 0 and corresponding Cauchy problem. Under the conditions: f( s) ∈ C^sup1 and satisfies (H)\qquad |f'(s)| ≤ C|s|^ϒ, 0 ≤ ϒ ≤ \frac{2}{n-2} if n ≥ 3; 0 ≤ ϒ < ∞ if n = 2 u_0(x) ∈ W^{2,p}(Ω) ∩ W^{2,2}(Ω) ∩ W^{1,p}_0(Ω)(W^{2,p}(R^n) ∩ W^{2,2}(R^n) for Cauchy problem), 2 ≤ p < ∞, we obtain the existence and uniqueness of global solution u(x, t) ∈ W^{1,∞}(0, T; W^{2,p}(Ω) ∩ W^{2,2}(Ω) ∩ W^{1,p}_0(Ω))(W^{1,∞}(0, T; W^{2,p}(R^n) ∩ W^{2,2} (R^n)) for Cauchy problem), so the results of [1] and [2] are generalized and improved in essential.  相似文献   

3.
ANECESSARYANDSUFFICIENTCONDITIONOFEXISTENCEOFGLOBALSOLUTIONSFORSOMENONLINEARHYPERBOLICEQUATIONS¥ZHANGQUANDE(DepartmentofMathe...  相似文献   

4.
In order better to research the singularities of the solutions $\[u \in H_{loc}^s(\Omega ),\Omega \subset {R^n},s > \frac{n}{2} + 1\]$ , for semilinear hyperbolic equations $\[u = f(u,Du)\]$, in this paper, a kind of weighted Sobolev space $\[({H^s})_{{P_\mu }}^\alpha \],\[\mu = 1,2,{p_1} = {D_i} - \left| {{D_x}} \right|,{P_2} = {D_i} + \left| {{D_x}} \right|\]$, closely related with the solutions of the equations, is presented. It is discussed that their products tacitly keep roughly $\[{H^{3x - n}}\]$ microlocal regularity on the characteristic directions for $\[{P_\mu }\]$ and invariance under nonlinear maps. Then it is obtained that roughly $\[{H^{3x - n}}\]$ propagation of singularities theorem is valid for $\[u = f(u)\]$.  相似文献   

5.
In this paper, we concern the existence of nontrivial ground state solutions of fractional $p$-Kirchhoff equation $$\left\{\begin{array}{ll} m\left(\|u\|^p\right) [(-\Delta)_p^su+V(x)|u|^{p-2}u] =f(x,u) \quad\text{in}\, \mathbb{R}^N, \vspace{0.2 cm}\\ \|u\|=\left(\int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy +\int_{\mathbb{R}^N}V(x)|u|^pdx\right)^{\frac{1}{p}}, \end{array}\right.$$ where $m:[0,+\infty)\rightarrow [0,+\infty)$ is a continuous function, $(-\Delta)_p^s$ is the fractional $p$-Laplacian operator with $0相似文献   

6.
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 $H¹_{loc}(\mathbb{H}^N)∩L^p_{loc}(H^N)$.  相似文献   

7.
We consider the Cauchy problem for the semilinear wave equation ${u_{tt} - \Delta u + V(x)u_t = |u|^p}$ .When ${V(x) = V_0(1 + |x|^2)^{-1/2}, V_0 \geq n}$ , we prove that the critical exponent for the problem is ${p_c(n)=\left\{\begin{array}{ll} 1+\frac{2}{n-1},& n \geq 2,\ +\infty,& n=1. \end{array}\right.}$   相似文献   

8.
The paper deal with the existence of positive solution for the following (p,q)-Laplacian nonlinear system \begin{align*} \left\{ \begin{array}{ll} -Δ_pu=a(x)(α_1f(v)+β_1h(u)), & x∈Ω,\\ -Δ_qv=b(x)(α_2g(u)+β_2k(v)),& x∈Ω,\\ u=v=0,& x∈∂Ω,\end{array} \right. \end{align*} where $Δ_p$ denotes the p-Laplacian operator defined by $Δ_{p}z=div(|∇_z|^{p-2}∇z), p>1, α_1, α_2, β_1, β_2$ are positive parameters and Ω is a bounded domain in $R^N(N > 1)$ with smooth boundary ∂Ω. Here a(x) and b(x) are $C^1$ sign-changing functions that maybe negative near the boundary and f, g, h, k are C^1 nondecreasing functions such that $f, g, h, k: [0,∞)→[0,∞); f (s), g(s), h(s), k(s) > 0; s > 0$ and $lim_{n→∞}\frac{f(Mg(x)^{\frac{1}{q-1}}}{x^{p-1}}=0$ for every $M > 0$. We discuss the existence of positive solution when $f, g, h, k, a(x)$ and $b(x)$ satisfy certain additional conditions. We use the method of sub-super solutions to establish our results.  相似文献   

9.
In this paper,\ we study fractional nonlinear Schrodinger equation (FNLS) with periodic boundary condition $$ \textbf{i}u_{t}=-(-\Delta)^{s_{0}} u-V*u-\epsilon f(x)|u|^4u,\ ~~x\in \mathbb{T}, ~~t\in \mathbb{R}, ~~s_{0}\in (\frac12,1),~~~~~~~~~~~~~~~~~~~~~~~~~~~~(0.1) $$ where $(-\Delta)^{s_{0}}$ is the Riesz fractional differentiation defined in [21] and $V*$ is the Fourier multiplier defined by $\widehat{V*u}(n)=V_n\widehat{u}(n),\ V_n\in\left[-1,1\right],$ and $f(x)$ is Gevrey smooth. We prove that for $0\leq|\epsilon|\ll1$ and appropriate $V$,\ the equation (0.1) admits a full dimensional KAM torus in the Gevrey space satisfying $ \frac12e^{-rn^{\theta}}\leq \left|q_n\right|\leq 2e^{-rn^{\theta}}, \theta\in (0,1),$ which generalizes the results given by [8-10] to fractional nonlinear Schrodinger equation.  相似文献   

10.
设$W_{\beta}(x)=\exp(-\frac{1}{2}|x|^{\beta})~(\beta > 7/6)$ 为Freud权, Freud正交多项式定义为满足下式$\int_{- \infty}^{\infty}p_{n}(x)p_{m}(x)W_{\beta}^{2}(x)\rd x=\left \{ \begin{array}{ll} 0 & \hspace{3mm} n \neq m , \\ 1 & \hspace{3mm}n = m \end{array} \right.$的  相似文献   

11.
This paper deals with the following IBV problem of nonlinear parabolic equation: $$\[\left\{ {\begin{array}{*{20}{c}} {{u_t} = \Delta u + F(u,{D_x}u,D_x^2u),(t,x) \in {B^ + } \times \Omega ,}\{u(0,x) = \varphi (x),x \in \Omega }\{u{|_{\partial \Omega }} = 0} \end{array}} \right.\]$$ where $\[\Omega \]$ is the exterior domain of a compact set in $\[{R^n}\]$ with smooth boundary and F satisfies $\[\left| {F(\lambda )} \right| = o({\left| \lambda \right|^2})\]$, near $\[\lambda = 0\]$. It is proved that when $\[n \ge 3\]$, under the suitable smoothness and compatibility conditions, the above problem has a unique global smooth solution for small initial data. Moreover, It is also proved that the solution has the decay property $\[{\left\| {u(t)} \right\|_{{L^\infty }(\Omega )}} = o({t^{ - \frac{n}{2}}})\]$, as $\[t \to + \infty \]$.  相似文献   

12.
奇异非线性$p-$调和方程的一类正整体解   总被引:2,自引:0,他引:2  
设p>1,β≥0是常数, n是自然数, 是一个连续函数.本文研究形如的奇异非线性p-调和方程的正整体解,给出了该类方程具有无穷多个其渐近阶刚好为|x|(2n-2)(当|x|→∞时)的径向对称的正整体解的若干充分条件.  相似文献   

13.
This paper studies the initial-boundary value problem of GBBM equations u_t - Δu_t = div f(u) \qquad\qquad\qquad(a) u(x, 0) = u_0(x)\qquad\qquad\qquad(b) u |∂Ω = 0 \qquad\qquad\qquad(c) in arbitrary dimensions, Ω ⊂ R^n. Suppose that. f(s) ∈ C¹ and |f'(s)| ≤ C (1+|s|^ϒ), 0 ≤ ϒ ≤ \frac{2}{n-2} if n ≥ 3, 0 ≤ ϒ < ∞ if n = 2, u_0 (x) ∈ W^{2⋅p}(Ω) ∩ W^{1⋅p}_0(Ω) (2 ≤ p < ∞), then ∀T > 0 there exists a unique global W^{2⋅p} solution u ∈ W^{1,∞}(0, T; W{2⋅p}(Ω)∩ W^{1⋅p}_0(Ω)), so the known results are generalized and improved essentially.  相似文献   

14.
In this paper, we are dedicated to studying the following singularly Choquard equation $$ -\varepsilon^2\Delta u+V(x)u=\varepsilon^{-\alpha}\left[I_{\alpha}\ast F(u)\right]f(u),\ \ \ \ x\in\R^2,$$ where $V(x)$ is a continuous real function on $\R^2$, $I_{\alpha}:\R^2\rightarrow\R$ is the Riesz potential, and $F$ is the primitive function of nonlinearity $f$ which has critical exponential growth. Using the Trudinger-Moser inequality and some delicate estimates, we show that the above problem admits at least one semiclassical ground state solution, for $\varepsilon>0$ small provided that $V(x)$ is periodic in $x$ or asymptotically linear as $|x|\rightarrow \infty$. In particular, a precise and fine lower bound of $\frac{f(t)}{e^{\beta_{0} t^{2}}}$ near infinity is introduced in this paper.  相似文献   

15.
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.  相似文献   

16.
设■该文主要讨论了上述奇异积分算子在广义的调幅空间上的有界性,其中粗糙核Ω∈L~1(S~(n-2))h(y)为有界的径向函数,而γ(y)是满足一定条件的超曲面.  相似文献   

17.
该文分析了四阶椭圆方程△u=|x|aup-1,x∈Ω; u=\Delta u=0 , x ∈аΩ, (Ω表示Rn中以原点为中心的球)基态解的集中性态,并证明了当p趋近于 2*=\frac{2n}{n-4} (n>4)时基态解up集中在Ω的边界附近.  相似文献   

18.
In this paper, we establish a singular Trudinger-Moser inequality for the whole hyperbolic space $$H^n: sup_{u∈W^{1,n}(H^n),∫_{H^n}|∇_H^nu|^ndμ ≤ 1}∫_{H^n}\frac{e^{α|u|\frac{n}{n-1}}-Σ^{n-2}_{k=0}\frac{α^k|u|^\frac{nk}{n-1}}{k!}}{ρ^β}dμ‹∞ ⇔ \frac{α}{α_n}+\frac{β}{n} ≤ 1,$$ where α>0,α ∈ [0,n), ρ and dμ are the distance function and volume element of $H^n$ respectively.  相似文献   

19.
In this paper, firstly, we investigate a class of singular eigenvalue problems with the perturbed Hardy–Sobolev operator, and obtain some properties of the eigenvalues and the eigenfunctions. (i.e. existence, simplicity, isolation and comparison results). Secondly, applying these properties of eigenvalue problem, and the linking theorem for two symmetric cones in Banach space, we discuss the following singular elliptic problem $$\left\{\begin{array}{ll}-\Delta_{p}u-a(x)\frac{|u|^{p-2}u}{|x|^{p}}= \lambda \eta(x)|u|^{p-2}u+ f(x,u) \quad x \in \Omega, \\ u =0 \quad\quad\quad\quad\quad\quad\quad x\in\partial \Omega, \end{array} \right.$$ where ${a(x)=(\frac{n-p}{p})^{p}q(x),}$ if 1 < p < n, ${a(x)=(\frac{n-1}{n})^{n} \frac{q(x)}{({\rm log}\frac{R}{|x|})^{n}},}$ if p = n, and prove the existence of a nontrivial weak solution for any ${\lambda \in \mathbb{R}.}$   相似文献   

20.
研究了$(n+p)$维双曲空间$\mathbb{H}^{n+p}$中完备非紧子流形的第一特征值的上界.特别地,证明了$\mathbb{H}^{n+p}$中具有平行平均曲率向量$H$和无迹第二基本形式有限$L^q(q\geq n)$范数的完备子流形的第一特征值不超过$\frac{(n-1)^2(1-|H|^2)}{4}$,和$\mathbb{H}^{n+1}(n\leq5)$中具有常平均曲率向量$H$和无迹第二基本形式有限$L^q(2(1-\sqrt{\frac{2}{n}})相似文献   

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