共查询到20条相似文献,搜索用时 31 毫秒
1.
We prove the monotonicity of nonnegative bounded solutions of the Dirichlet problem for the quasilinear elliptic equation ?Δpu = f(u), p ≥ 3, in a half-space. This assertion implies new results on the nonexistence of solutions for the case in which f(u) = uq with appropriate values of q. 相似文献
2.
Teng Fei Zhao 《数学学报(英文版)》2017,33(7):911-925
We consider the defocusing nonlinear Schr?dinger equations iu_t +△u =|u|~(p_u) with p being an even integer in dimensions d≥ 5. We prove that an a priori bound of critical norm implies global well-posedness and scattering for the solution. 相似文献
3.
We study the existence, uniqueness, and asymptotic behavior of blow-up solutions for a general quasilinear elliptic equation of the type ?Δ p u = a(x)u m ?b(x)f(u) with p > 1 and 0 < m < p?1. The main technical tool is a new comparison principle that enables us to extend arguments for semilinear equations to quasilinear ones. Indeed, this paper is an attempt to generalize all available results for the semilinear case with p = 2 to the quasilinear case with p > 1. 相似文献
4.
In this work we study the existence and regularity of solutions of the equation Δ p 2 u = λm|u| q?2 u with the boundary conditions of Navier in the case p ≠ q. 相似文献
5.
E. K. Kostousova 《Proceedings of the Steklov Institute of Mathematics》2016,293(1):140-150
Based on the method of nonlinear capacity, we study the nonexistence of nonnegative monotonic solutions for the quasilinear elliptic inequality of the form ?Δpu ≥ uq in a half-space in terms of the parameters p and q. 相似文献
6.
In this paper, we study the nonexistence and longtime behavior of weak solution for the degenerate parabolic equation ? t u n = u m div(|?u m | p?2?u m ) + γ|?u m | p + β u n with zero boundary condition. Blow-up time is derived when the blow-up does occur. 相似文献
7.
Let p be a prime greater than five and A the mod p Steenrod algebra. In this paper, we prove that \(h_n h_m \tilde \delta _{s + 4} \in Ext_A^{s + 6,t(s,n,m) + s} (Z/p,Z/p)\) is nontrivial in the Adams E2-term when m ≥ n + 2 ≥ 7 and 0 ≤ s < p ? 4, and trivial in the Adams E2-term when m ≥ n + 2 = 6 and 0 ≤ s < p ? 4, where \(\tilde \delta _{s + 4} \) stands for the fourth Greek letter element and t(s, n, m) = 2(p ? 1)[(s + 1) + (s + 2)p + (s + 3)p2 + (s + 4)p3 + pn + pm]. 相似文献
8.
A. I. Aristov 《Differential Equations》2018,54(9):1137-1146
The equation ?2u/?t?x + up?u/?x = uq describing a nonstationary process in semiconductors, with parameters p and q that are a nonnegative integer and a positive integer, respectively, and satisfy p + q ≥ 2, is considered in the half-plane (x, t) ∈ ? × (0,∞). All in all, fourteen families of its exact solutions are constructed for various parameter values, and qualitative properties of these solutions are noted. One of these families is defined for all parameter values indicated above. 相似文献
9.
Based on the eigensystem {λj,φj} of -Δ, the multiple solutions for nonlinear problem Δu + f(u) = 0 in Ω,u = 0 on ?Ω are approximated. A new search-extension method (SEM), which consists of three steps in three level subspaces, is proposed. Numerical simulations for several typical nonlinear cases, i.e. f(u) = u 3, u 2, (u - p), u 2(u 2 - p), are completed and some conjectures are presented. 相似文献
10.
V. A. Sharafutdinov 《Siberian Mathematical Journal》2018,59(4):731-752
For a real solution (u, p) to the Euler stationary equations for an ideal fluid, we derive an infinite series of the orthogonality relations that equate some linear combinations of mth degree integral momenta of the functions uiuj and p to zero (m = 0, 1,... ). In particular, the zeroth degree orthogonality relations state that the components ui of the velocity field are L2-orthogonal to each other and have coincident L2-norms. Orthogonality relations of degree m are valid for a solution belonging to a weighted Sobolev space with the weight depending on m. 相似文献
11.
We investigate the problem (P λ) ?Δu = λb(x)|u| q?2 u + a(x)|u| p?2 u in Ω, ?u/?n = 0 on ?Ω, where Ω is a bounded smooth domain in R N (N ≥ 2), 1 < q < 2 < p, λ ∈ R, and a, b ∈ \({C^\alpha }\left( {\overline \Omega } \right)\) with 0 < α < 1. Under certain indefinite type conditions on a and b, we prove the existence of two nontrivial nonnegative solutions for small |λ|. We then characterize the asymptotic profiles of these solutions as λ → 0, which in some cases implies the positivity and ordering of these solutions. In addition, this asymptotic analysis suggests the existence of a loop type component in the non-negative solutions set. We prove the existence of such a component in certain cases, via a bifurcation and a topological analysis of a regularized version of (P λ). 相似文献
12.
M. V. Voitovich 《Mathematical Notes》2016,99(5-6):840-850
In the present paper, a 2mth-order quasilinear divergence equation is considered under the condition that its coefficients satisfy the Carathéodory condition and the standard conditions of growth and coercivity in the Sobolev space Wm,p(Ω), Ω ? Rn, p > 1. It is proved that an arbitrary generalized (in the sense of distributions) solution u ∈ W0m,p (Ω) of this equation is bounded if m ≥ 2, n = mp, and the right-hand side of this equation belongs to the Orlicz–Zygmund space L(log L)n?1(Ω). 相似文献
13.
Qing Guo 《应用数学学报(英文版)》2016,32(1):137-162
We investigate the nonlinear Schrödinger equation iu t +Δu+|u| p?1 u = 0with 1+ 4/N < p < 1+ 4/N?2 (when N = 1, 2, 1 + 4/N < p < ∞) in energy space H 1 and study the divergent property of infinite-variance and nonradial solutions. If \(M{\left( u \right)^{\frac{{1 - {s_C}}}{{{s_C}}}}}E\left( u \right) \prec M{\left( Q \right)^{\frac{{1 - {s_C}}}{{{s_C}}}}}E\left( Q \right)\) and \(\left\| {{u_0}} \right\|_2^{\frac{{1 - {s_c}}}{{{s_c}}}}\left\| {\nabla {u_0}} \right\|_2^{\frac{{1 - {s_c}}}{{{s_c}}}}{\left\| {\nabla Q} \right\|_2}\), then either u(t) blows up in finite forward time or u(t) exists globally for positive time and there exists a time sequence t n → +∞ such that \({\left\| {\nabla u\left( {{t_n}} \right)} \right\|_2} \to + \infty \). Here Q is the ground state solution of ?(1?s c )Q+ΔQ+Q p?1 Q = 0. A similar result holds for negative time. This extend the result of the 3D cubic Schrödinger equation obtained by Holmer to the general mass-supercritical and energy-subcritical case. 相似文献
14.
We investigate the nonnegative solutions of the system involving the fractional Laplacian: where 0 < α < 1, n > 2, f i (u), 1 ≤ i ≤ m, are real-valued nonnegative functions of homogeneous degree p i ≥ 0 and nondecreasing with respect to the independent variables u 1, u 2,..., u m . By the method of moving planes, we show that under the above conditions, all the positive solutions are radially symmetric and monotone decreasing about some point x 0 if p i = (n + 2α)/(n ? 2α) for each 1 ≤ i ≤ m; and the only nonnegative solution of this system is u ≡ 0 if 1 < p i < (n + 2α)/(n ? 2α) for all 1 ≤ i ≤ m.
相似文献
$$\left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {( - \Delta )^\alpha u_i (x) = f_i (u),} & {x \in \mathbb{R}^n , i = 1,2, \ldots ,m,} \\ \end{array} } \\ {u(x) = (u_1 (x),u_2 (x), \ldots ,u_m (x)),} \\ \end{array} } \right.$$
15.
Ji-xiu Wang 《应用数学学报(英文版)》2017,33(2):417-434
In this paper, we study the existence of semiclassical states for some p-Laplacian equation. Under given conditions and minimax methods, we show that this problem has at least one positive solution provided that ε ≤ E; for any m ∈ ?, it has m pairs solutions if ε ≤ E m , where E, Em are sufficiently small positive numbers. Moreover, these solutions are closed to zero in W1,p(? N ) as ε → 0. 相似文献
16.
This paper studies heat equation with variable exponent u t = Δu + up(x) + u q in ? N × (0, T), where p(x) is a nonnegative continuous, bounded function, 0 < p? = inf p(x) ≤ p(x) ≤ sup p(x) = p+. It is easy to understand for the problem that all nontrivial nonnegative solutions must be global if and only if max {p+, q} ≤ 1. Based on the interaction between the two sources with fixed and variable exponents in the model, some Fujita type conditions are determined that that all nontrivial nonnegative solutions blow up in finite time if 0 < q ≤ 1 with p+ > 1, or 1 < q < 1 + \(\frac{2}{N}\). In addition, if q > 1 + \(\frac{2}{N}\), then (i) all solutions blow up in finite time with 0 < p? ≤ p+ ≤ 1 + \(\frac{2}{N}\); (ii) there are both global and nonglobal solutions for p? > 1 + \(\frac{2}{N}\); and (iii) there are functions p(x) such that all solutions blow up in finite time, and also functions p(x) such that the problem possesses global solutions when p? < 1 + \(\frac{2}{N}\) < p+. 相似文献
17.
M. D. Surnachev 《Moscow University Mathematics Bulletin》2009,64(2):67-69
The semilinear equation Δu = |u|σ?1 u is considered in the exterior of a ball in ? n , n ≥ 3. It is shown that if the exponent σ is greater than a “critical” value (= n/n?2), then for x → ∞ the leading term of the asymptotics of any solution is a linear combination of derivatives of the fundamental solution. It is shown that there exist solutions with the indicated leading term of an asymptotics of such a type. 相似文献
18.
Changchun Liu 《Monatshefte für Mathematik》2012,165(2):237-249
In this paper, we study the initial-boundary value problem of porous medium equation ρ(x)u t = Δu m + V(x)h(t)u p in a cone D = (0, ∞) × Ω, where \({V(x)\,{\sim}\, |x|^\sigma, h(t)\,{\sim}\, t^s}\). Let ω 1 denote the smallest Dirichlet eigenvalue for the Laplace-Beltrami operator on Ω and let l denote the positive root of l 2 + (n ? 2)l = ω 1. We prove that if \({m < p \leq 1+(m-1)(1+s)+\frac{2(s+1)+\sigma}{n+l}}\), then the problem has no global nonnegative solutions for any nonnegative u 0 unless u 0 = 0; if \({p >1 +(m-1)(1+s)+\frac{2(s+1)+\sigma}{n+l}}\), then the problem has global solutions for some u 0 ≥ 0. 相似文献
19.
We investigate the non-homogeneous modular Dirichlet problem Δ p (·)u(x) = f (x) (where Δ p (·)u(x) = div(|?u|p(x-2)?u(x)) from the functional analytic point of view and we prove the stability of the solutions \({\left( {{u_{{p_i}}}} \right)_i}\) of the equation \({\Delta _{{p_i}\left( \cdot \right)}}{u_{{p_i}\left( \cdot \right)}} = f\) as p i (·) → q(·) via Gamma-convergence of sequence of appropriate functionals. 相似文献
20.
The Shannon complexity of a function system over a q-element finite field which contains m functions of n variables in the class of polarized polynomial forms is exactly evaluated: L q PPF (n,m) = q n for all n ≥ 1, m ≥ 2, and all possible odd q. It has previously been known that L2PPF (n,m) = 2 n and L3PPF (n,m) = 3 n for all n ≥ 1 and m ≥ 2. 相似文献