共查询到20条相似文献,搜索用时 36 毫秒
1.
C.A. Santos 《Nonlinear Analysis: Theory, Methods & Applications》2009,71(12):6038-6043
We establish a result concerning the existence of entire, positive, classical and bounded solutions which converge to zero at infinity for the semi-linear equation −Δu=λf(x,u),x∈RN, where f:RN×(0,∞)→[0,∞) is a suitable function and λ>0 is a real parameter. This result completes the principal theorem of A. Mohammed [A. Mohammed, Ground state solutions for singular semi-linear elliptic equations, Nonlinear Analysis (2008) doi:10.1016/j.na.2008.11.080] mainly because his result does not address the super-linear terms at infinity. Penalty arguments, lower-upper solutions and an approximation procedure will be used. 相似文献
2.
We consider the entire graph S of a continuous real function over RN−1 with N?3. Let Ω be a domain in RN with S as a boundary. Consider in Ω the heat flow with initial temperature 0 and boundary temperature 1. The problem we consider is to characterize S in such a way that there exists a stationary isothermic surface in Ω. We show that S must be a hyperplane under some general conditions on S. This is related to Liouville or Bernstein-type theorems for some elliptic Monge-Ampère-type equation. 相似文献
3.
Zhonghai Xu Zhenguo FengJiashan Zheng 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(1):1-8
The Keldysh equation is a more general form of the classic Tricomi equation from fluid dynamics. Its well-posedness and the regularity of its solution are interesting and important. The Keldysh equation is elliptic in y>0 and is degenerate at the line y=0 in R2. Adding a special nonlinear absorption term, we study a nonlinear degenerate elliptic equation with mixed boundary conditions in a piecewise smooth domain—similar to the potential fluid shock reflection problem. By means of an elliptic regularization technique, a delicate a priori estimate and compact argument, we show that the solution of a mixed boundary value problem of the Keldysh equation is smooth in the interior and Lipschitz continuous up to the degenerate boundary under some conditions. We believe that this kind of regularity result for the solution will be rather useful. 相似文献
4.
Yanping Cao 《Journal of Differential Equations》2006,231(2):755-767
We show that the only locally integrable stationary solutions to the integrated Kuramoto-Sivashinsky equation in R and R2 are the trivial constant solutions. We extend our technique and prove similar results to other nonlinear elliptic problems in RN. 相似文献
5.
We construct positive solutions of the semilinear elliptic problem with Dirichet boundary conditions, in a bounded smooth domain Ω⊂RN(N?4), when the exponent p is supercritical and close enough to and the parameter λ∈R is small enough. As , the solutions have multiple blow up at finitely many points which are the critical points of a function whose definition involves Green's function. Our result extends the result of Del Pino et al. (J. Differential Equations 193(2) (2003) 280) when Ω is a ball and the solutions are radially symmetric. 相似文献
6.
Solutions concentrating on higher dimensional subsets for singularly perturbed elliptic equations II
We construct spike layered solutions for the semilinear elliptic equation −ε2Δu+V(x)u=K(x)up−1 on a domain Ω⊂RN which may be bounded or unbounded. The solutions concentrate simultaneously on a finite number of m-dimensional spheres in Ω. These spheres accumulate as ε→0 at a prescribed sphere in Ω whose location is determined by the potential functions V,K. 相似文献
7.
Shihui Zhu Jian ZhangHan Yang 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(17):6244-6255
This paper is concerned with the Cauchy problem for the biharmonic nonlinear Schrödinger equation with L2-super-critical nonlinearity. By establishing the profile decomposition of bounded sequences in H2(RN), the best constant of a Gagliardo-Nirenberg inequality is obtained. Moreover, a sufficient condition for the global existence of the solution to the biharmonic nonlinear Schrödinger equation is given. 相似文献
8.
Elves A.B. Silva Gilberto F. Vieira 《Nonlinear Analysis: Theory, Methods & Applications》2010,72(6):2935-2949
In this article the existence of one solution for a class of asymptotically periodic equations in the euclidean space is established. The basic tools employed here are the Mountain Pass Theorem and the Concentration-Compactness Principle. By using a change of variable, the quasilinear equation is reduced to a semilinear equation, whose respective associated functional is well defined in H1(RN) and satisfies the geometric hypotheses of the Mountain Pass Theorem. 相似文献
9.
The existence and concentration behavior of nodal solutions are established for the equation −?2Δu+V(z)u=f(u) in Ω, where Ω is a domain in R2, not necessarily bounded, V is a positive Hölder continuous function and f∈C1 is an odd function having critical exponential growth. 相似文献
10.
Zhijun Zhang 《Nonlinear Analysis: Theory, Methods & Applications》2010,73(10):3348-3363
In this paper, we study the boundary behavior of solutions to boundary blow-up elliptic problems , where Ω is a bounded domain with smooth boundary in RN, q>0, , which is positive in Ω and may be vanishing on the boundary and rapidly varying near the boundary, and f is rapidly varying or normalized regularly varying at infinity. 相似文献
11.
Catherine Bandle 《Journal of Differential Equations》2011,251(8):2143-827
It is well known that the heat kernel in the hyperbolic space has a different behavior for large times than the one in the Euclidean space. The main purpose of this paper is to study its effect on the positive solutions of Cauchy problems with power nonlinearities. Existence and non-existence results for local solutions are derived. Emphasis is put on their long time behavior and on Fujita?s phenomenon. To have the same situation as for the Cauchy problem in RN, namely finite time blow up for all solutions if the exponent is smaller than a critical value and existence of global solutions only for powers above the critical exponent, we must introduce a weight depending exponentially on the time. In this respect the situation is similar to problems in bounded domains with Dirichlet boundary conditions. Important tools are estimates for the heat kernel in the hyperbolic space and comparison principles. 相似文献
12.
Fethi Mahmoudi 《Advances in Mathematics》2007,209(2):460-525
We consider the equation −ε2Δu+u=up in Ω⊆RN, where Ω is open, smooth and bounded, and we prove concentration of solutions along k-dimensional minimal submanifolds of ∂Ω, for N?3 and for k∈{1,…,N−2}. We impose Neumann boundary conditions, assuming 1<p<(N−k+2)/(N−k−2) and ε→0+. This result settles in full generality a phenomenon previously considered only in the particular case N=3 and k=1. 相似文献
13.
We consider the boundary value problem Δu+up=0 in a bounded, smooth domain Ω in R2 with homogeneous Dirichlet boundary condition and p a large exponent. We find topological conditions on Ω which ensure the existence of a positive solution up concentrating at exactly m points as p→∞. In particular, for a nonsimply connected domain such a solution exists for any given m?1. 相似文献
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16.
Yihong Du 《Journal of Differential Equations》2008,244(1):117-169
Let u? be a single layered radially symmetric unstable solution of the Allen-Cahn equation −?2Δu=u(u−a(|x|))(1−u) over the unit ball with Neumann boundary conditions. Based on our estimate of the small eigenvalues of the linearized eigenvalue problem at u? when ? is small, we construct solutions of the form u?+v?, with v? non-radially symmetric and close to zero in the unit ball except near one point x0 such that |x0| is close to a nondegenerate critical point of a(r). Such a solution has a sharp layer as well as a spike. 相似文献
17.
Jan W. Cholewa 《Journal of Differential Equations》2010,249(3):485-588
We consider monotone semigroups in ordered spaces and give general results concerning the existence of extremal equilibria and global attractors. We then show some applications of the abstract scheme to various evolutionary problems, from ODEs and retarded functional differential equations to parabolic and hyperbolic PDEs. In particular, we exhibit the dynamical properties of semigroups defined by semilinear parabolic equations in RN with nonlinearities depending on the gradient of the solution. We consider as well systems of reaction-diffusion equations in RN and provide some results concerning extremal equilibria of the semigroups corresponding to damped wave problems in bounded domains or in RN. We further discuss some nonlocal and quasilinear problems, as well as the fourth order Cahn-Hilliard equation. 相似文献
18.
Ryuji Kajikiya 《Nonlinear Analysis: Theory, Methods & Applications》2010,73(7):2117-2131
In this paper, a superlinear elliptic equation whose coefficient diverges on the boundary is studied in any bounded domain Ω under the zero Dirichlet boundary condition. Although the equation has a singularity on the boundary, a solution is smooth on the closure of the domain. Indeed, it is proved that the problem has a positive solution and infinitely many solutions without positivity, which belong to or . Moreover, it is proved that a positive solution has a higher order regularity up to . 相似文献
19.
Michael E. Filippakis 《Journal of Differential Equations》2008,245(7):1883-1922
We consider a nonlinear elliptic equation driven by the p-Laplacian with Dirichlet boundary conditions. Using variational techniques combined with the method of upper-lower solutions and suitable truncation arguments, we establish the existence of at least five nontrivial solutions. Two positive, two negative and a nodal (sign-changing) solution. Our framework of analysis incorporates both coercive and p-superlinear problems. Also the result on multiple constant sign solutions incorporates the case of concave-convex nonlinearities. 相似文献
20.
Norimichi Hirano 《Journal of Differential Equations》2009,247(5):1311-2003
Let N?3, 2*=2N/(N−2) and Ω⊂RN be a bounded domain with a smooth boundary ∂Ω and 0∈Ω. Our purpose in this paper is to consider the existence of solutions of Hénon equation: