共查询到20条相似文献,搜索用时 15 毫秒
1.
Liang Zhao 《偏微分方程(英文版)》2012,25(1):90-102
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.
V. M. Babich 《Journal of Mathematical Sciences》1985,28(5):628-632
One considers the problem of the asymptotic behavior for K→+∞ of the solution of the Cauchy problem $$u_{tt} - u_{xx} + \kappa ^2 u = 0; u|_{t = 0} = \theta (x), u_t |_{t = 0} = 0 (t > 0 - fixed)$$ Hereθ(x) is the Heaviside function. In the neighborhood of the characteristics x=±t function u(x,t)?2 oscillates exceptionally fast (the wavelength is of order k?2). Near the t axis the asymptotics of u(x,t) contains the Fresnel integral. 相似文献
3.
Mohamed Berbiche & Ali Hakem 《偏微分方程(英文版)》2012,25(1):1-20
We considered the Cauchy problem for the fractional wave-diffusion equation $$D^αu-Δ|u|^{m-1}u+(-Δ)^{β/2}D^γ|u|^{l-1}u=h(x,t)|u|^p+f(x,t)$$ with given initial data and where p > 1, 1 < α < 2, 0 < β < 2, 0 < γ < 1. Nonexistence results and necessary conditions for global existence are established by means of the test function method. This results extend previous works. 相似文献
4.
The initial boundary value problem
$ {*{20}{c}} {\rho {u_{tt}} - {{\left( {\Gamma {u_x}} \right)}_x} + A{u_x} + Bu = 0,} \hfill & {x > 0,\quad 0 < t < T,} \hfill \\ {u\left| {_{t = 0}} \right. = {u_t}\left| {_{t = 0}} \right. = 0,} \hfill & {x \geq 0,} \hfill \\ {u\left| {_{x = 0}} \right. = f,} \hfill & {0 \leq t \leq T,} \hfill \\ $ \begin{array}{*{20}{c}} {\rho {u_{tt}} - {{\left( {\Gamma {u_x}} \right)}_x} + A{u_x} + Bu = 0,} \hfill & {x > 0,\quad 0 < t < T,} \hfill \\ {u\left| {_{t = 0}} \right. = {u_t}\left| {_{t = 0}} \right. = 0,} \hfill & {x \geq 0,} \hfill \\ {u\left| {_{x = 0}} \right. = f,} \hfill & {0 \leq t \leq T,} \hfill \\ \end{array} 相似文献
5.
We consider the singularly perturbed quasilinear Dirichlet problems of the form {-∈Δ_pu = f(u) in Ω u ≥ 0 in , u = 0 on ∂ Ω where Δ_pu = div(|Du|^{p-2}Du), p > 1, f is subcritical. ∈ > 0 is a small parameter and is a bounded smooth domain in R^N (N ≥ 2). When Ω = B_1 = {x; |x| < 1} is the unit ball, we show that the least energy solution is radially symmetric, the solution is also unique and has a unique peak point at origin as ∈ → 0. 相似文献
6.
We study the existence of weak solutions for a nonlinear elliptic system of Lane-Emden type $$\left\{\begin{array}{ll} -\Delta u \; = \; {\rm sgn}(v)|v|^{p-1} & {\rm in}\;\mathbb{R}^N, \\ -\Delta v \; = \; -\rho(x){\rm sgn}(u)|u|^{\frac{1}{p-1}} + f(x, u) & {\rm in}\;\mathbb{R}^N, \\ u, v \to 0 \quad {\rm as} \quad |x| \to +\infty, \end{array}\right.$$ by means of the Mountain Pass Theorem and some compact imbeddings in weighted Sobolev spaces. 相似文献
7.
H. G. Ghazaryan V. N. Margaryan 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2010,45(4):239-249
It is proved that if P(D) is a regular, almost hypoelliptic operator and
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