共查询到20条相似文献,搜索用时 93 毫秒
1.
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
2.
Shingo Takeuchi 《Journal of Differential Equations》2011,251(8):2196-2208
This paper concerns the formation of a coincidence set for the positive solution of the boundary value problem: −εΔpu=uq−1f(a(x)−u) in Ω with u=0 on ∂Ω, where ε is a positive parameter, Δpu=div(|∇u|p−2∇u), 1<q?p<∞, f(s)∼|s|θ−1s(s→0) for some θ>0 and a(x) is a positive smooth function satisfying Δpa=0 in Ω with infΩ|∇a|>0. It is proved in this paper that if 0<θ<1 the coincidence set Oε={x∈Ω:uε(x)=a(x)} has a positive measure for small ε and converges to Ω with order O(ε1/p) as ε→0. Moreover, it is also shown that if θ?1, then Oε is empty for any ε>0. The proofs rely on comparison theorems and the energy method for obtaining local comparison functions. 相似文献
3.
Alfredo Cano 《Journal of Differential Equations》2007,237(1):133-158
We consider the problem −Δu+a(x)u=f(x)|u|2*−2u in Ω, u=0 on ∂Ω, where Ω is a bounded smooth domain in RN, N?4, is the critical Sobolev exponent, and a,f are continuous functions. We assume that Ω, a and f are invariant under the action of a group of orthogonal transformations. We obtain multiplicity results which contain information about the symmetry and symmetry-breaking properties of the solutions, and about their nodal domains. Our results include new multiplicity results for the Brezis-Nirenberg problem −Δu+λu=|u|2*−2u in Ω, u=0 on ∂Ω. 相似文献
4.
In this paper we consider radially symmetric solutions of the nonlinear Dirichlet problem Δu+f(|x|,u)=0 in Ω, where Ω is a ball in RN, N?3 and f satisfies some appropriate assumptions. We prove existence of radially symmetric solutions with k prescribed number of zeros. Moreover, when f(|x|,u)=K(|x|)|u|p−1u, using the uniqueness result due to Tanaka (2008) [21], we verify that these solutions are non-degenerate and we prove that their radial Morse index is exactly k. 相似文献
5.
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
6.
Frédéric Robert 《Journal of Differential Equations》2006,231(1):135-164
We let Ω be a smooth bounded domain of R4 and a sequence of functions (Vk)k∈N∈C0(Ω) such that limk→+∞Vk=1 in . We consider a sequence of functions (uk)k∈N∈C4(Ω) such that Δ2uk=Vke4uk in Ω for all k∈N. We address in this paper the question of the asymptotic behavior of the (uk)'s when k→+∞. The corresponding problem in dimension 2 was considered by Brézis and Merle, and Li and Shafrir (among others), where a blow-up phenomenon was described and where a quantization of this blow-up was proved. Surprisingly, as shown by Adimurthi, Struwe and the author in [Adimurthi, F. Robert and M. Struwe, Concentration phenomena for Liouville equations in dimension four, J. Eur. Math. Soc., in press, available on http://www-math.unice.fr/~frobert], a similar quantization phenomenon does not hold for this fourth-order problem. Assuming that the uk's are radially symmetrical, we push further the analysis of the mentioned work. We prove that there are exactly three types of blow-up and we describe each type in a very detailed way. 相似文献
7.
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
8.
In this paper we study existence and properties of solutions of the problem Δw=0 on the half-space with nonlinear boundary condition ∂w/∂η+w=|w|p−2w where 2<p<2(N−1)/(N−2) and N?3. We obtain a ground state solution w=w(x1,…,xN−1,t) which is radial and has exponential decay in the first N−1 variables. Moreover, w has sharp polynomial decay in the variable t. 相似文献
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10.
For the equation −Δu=||xα|−2|up−1, 1<|x|<3, we prove the existence of two solutions for α large, and of two additional solutions when p is close to the critical Sobolev exponent 2∗=2N/(N−2). A symmetry-breaking phenomenon appears, showing that the least-energy solutions cannot be radial functions. 相似文献
11.
Let Ω be a bounded domain in R2, u+=u if u?0, u+=0 if u<0, u−=u+−u. In this paper we study the existence of solutions to the following problem arising in the study of a simple model of a confined plasma
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Mihai Mih?ilescu 《Journal of Mathematical Analysis and Applications》2007,330(1):416-432
We study the boundary value problem −div(log(1+q|∇u|)|∇u|p−2∇u)=f(u) in Ω, u=0 on ∂Ω, where Ω is a bounded domain in RN with smooth boundary. We distinguish the cases where either f(u)=−λ|u|p−2u+|u|r−2u or f(u)=λ|u|p−2u−|u|r−2u, with p, q>1, p+q<min{N,r}, and r<(Np−N+p)/(N−p). In the first case we show the existence of infinitely many weak solutions for any λ>0. In the second case we prove the existence of a nontrivial weak solution if λ is sufficiently large. Our approach relies on adequate variational methods in Orlicz-Sobolev spaces. 相似文献
15.
Starting with the famous article [A. Gidas, W.M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979) 209-243], many papers have been devoted to the uniqueness question for positive solutions of −Δu=λu+up in Ω, u=0 on ∂Ω, where p>1 and λ ranges between 0 and the first Dirichlet eigenvalue λ1(Ω) of −Δ. For the case when Ω is a ball, uniqueness could be proved, mainly by ODE techniques. But very little is known when Ω is not a ball, and then only for λ=0. In this article, we prove uniqueness, for all λ∈[0,λ1(Ω)), in the case Ω=2(0,1) and p=2. This constitutes the first positive answer to the uniqueness question in a domain different from a ball. Our proof makes heavy use of computer assistance: we compute a branch of approximate solutions and prove existence of a true solution branch close to it, using fixed point techniques. By eigenvalue enclosure methods, and an additional analytical argument for λ close to λ1(Ω), we deduce the non-degeneracy of all solutions along this branch, whence uniqueness follows from the known bifurcation structure of the problem. 相似文献
16.
In this paper, we study the existence of multiple positive solutions to some Hamiltonian elliptic systems −Δv=λu+up+εf(x), −Δu=μv+vq+δg(x) in Ω;u,v>0 in Ω; u=v=0 on ∂Ω, where Ω is a bounded domain in RN (N?3); 0?f, g∈L∞(Ω); 1/(p+1)+1/(q+1)=(N−2)/N, p,q>1; λ,μ>0. Using sub- and supersolution method and based on an adaptation of the dual variational approach, we prove the existence of at least two nontrivial positive solutions for all λ,μ∈(0,λ1) and ε,δ∈(0,δ0), where λ1 is the first eigenvalue of the Laplace operator −Δ with zero Dirichlet boundary conditions and δ0 is a positive number. 相似文献
17.
Louis Dupaigne 《Journal de Mathématiques Pures et Appliquées》2007,87(6):563-581
We are concerned with singular elliptic problems of the form −Δu±p(d(x))g(u)=λf(x,u)+μa|∇u| in Ω, where Ω is a smooth bounded domain in RN, d(x)=dist(x,∂Ω), λ>0, μ∈R, 0<a?2, and f is a nondecreasing function. We assume that p(d(x)) is a positive weight with possible singular behavior on the boundary of Ω and that the nonlinearity g is unbounded around the origin. Taking into account the competition between the anisotropic potential p(d(x)), the convection term a|∇u|, and the singular nonlinearity g, we establish various existence and nonexistence results. 相似文献
18.
Marcelo Montenegro Olivâine Santana de Queiroz 《Journal of Differential Equations》2009,246(2):482-511
We study the nonlinear elliptic problem −Δu=χ{u>0}(logu+λf(x,u)) in Ω⊂Rn with u=0 on ∂Ω. The function is nondecreasing, sublinear and fu is continuous. For every λ>0, we obtain a maximal solution uλ?0 and prove its global regularity . There is a constant λ∗ such that uλ vanishes on a set of positive measure for 0<λ<λ∗, and uλ>0 for λ>λ∗. If f is concave, for λ>λ∗ we characterize uλ by its stability. 相似文献
19.
We study the boundary value problems for Monge-Ampère equations: detD2u=e−u in Ω⊂Rn, n?1, u|∂Ω=0. First we prove that any solution on the ball is radially symmetric by the argument of moving plane. Then we show there exists a critical radius such that if the radius of a ball is smaller than this critical value there exists a solution, and vice versa. Using the comparison between domains we can prove that this phenomenon occurs for every domain. Finally we consider an equivalent problem with a parameter detD2u=e−tu in Ω, u|∂Ω=0, t?0. By using Lyapunov-Schmidt reduction method we get the local structure of the solutions near a degenerate point; by Leray-Schauder degree theory, a priori estimate and bifurcation theory we get the global structure. 相似文献
20.
Weilin Zou 《Nonlinear Analysis: Theory, Methods & Applications》2010,73(9):3069-3082
This paper deals with a class of degenerate quasilinear elliptic equations of the form −div(a(x,u,∇u)=g−div(f), where a(x,u,∇u) is allowed to be degenerate with the unknown u. We prove existence of bounded solutions under some hypothesis on f and g. Moreover we prove that there exists a renormalized solution in the case where g∈L1(Ω) and f∈(Lp′(Ω))N. 相似文献