共查询到20条相似文献,搜索用时 968 毫秒
1.
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
2.
We prove regularity and partial regularity results for finite Morse index solutions u∈H1(Ω)∩Lp(Ω) to the Lane-Emden equation −Δu=|u|p−1u in Ω. 相似文献
3.
《Nonlinear Analysis: Theory, Methods & Applications》2004,57(3):349-362
We show that in a smooth bounded domain , n⩾2, all global nonnegative solutions of ut−Δum=up with zero boundary data are uniformly bounded in by a constant depending on and τ but not on u0, provided that 1<m<p<[(n+1)/(n−1)]m. Furthermore, we prove an a priori bound in depending on under the optimal condition 1<m<p<[(n+2)/(n−2)]m. 相似文献
4.
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
5.
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
6.
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
7.
Vitali Liskevich I.I. Skrypnik 《Journal of Mathematical Analysis and Applications》2008,338(1):536-544
We study the problem of removability of isolated singularities for a general second-order quasi-linear equation in divergence form −divA(x,u,∇u)+a0(x,u)+g(x,u)=0 in a punctured domain Ω?{0}, where Ω is a domain in Rn, n?3. The model example is the equation −Δpu+gu|u|p−2+u|u|q−1=0, q>p−1>0, p<n. Assuming that the lower-order terms satisfy certain non-linear Kato-type conditions, we prove that for all point singularities of the above equation are removable, thus extending the seminal result of Brezis and Véron. 相似文献
8.
《Comptes Rendus de l'Academie des Sciences Series IIA Earth and Planetary Science》1997,324(2):165-168
Due to the compactness of the operator (-Δp)1 Nƒ, 1 < p < ∞, where Δp is the p-Laplacian and Nƒ is the Nemytskii operator corresponding to a Caratheodory function ƒ: Ω × R → R, which satisfies a particular growth condition, the homotopy invariance of Leray-Schauder degree can be used in order to prove the existence of a W01,p (Ω)-solution for the equation -Δp u = Nƒ u. 相似文献
9.
Let Ω⊂Rn be a bounded Lipschitz domain with a cone-like corner at 0∈∂Ω. We prove existence of at least two positive unbounded very weak solutions of the problem −Δu=up in Ω, u=0 on ∂Ω, which have a singularity at 0, for any p slightly bigger that the generalized Brezis-Turner exponent p*. On an example of a planar polygonal domain the actual size of the p-interval on which the existence result holds is computed. The solutions are found variationally as perturbations of explicitly constructed singular solutions in cones. This approach also makes it possible to find numerical approximations of the two very weak solutions on Ω following a gradient flow of a suitable functional and using the mountain pass algorithm. Two-dimensional examples are presented. 相似文献
10.
M. Loayza 《Journal of Differential Equations》2006,229(2):509-528
We study the existence, uniqueness and regularity of positive solutions of the parabolic equation ut−Δu=a(x)uq+b(x)up in a bounded domain and with Dirichlet's condition on the boundary. We consider here a∈Lα(Ω), b∈Lβ(Ω) and 0<q?1<p. The initial data u(0)=u0 is considered in the space Lr(Ω), r?1. In the main result (0<q<1), we assume a,b?0 a.e. in Ω and we assume that u0?γdΩ for some γ>0. We find a unique solution in the space . 相似文献
11.
Ahmed Hamydy 《Journal of Mathematical Analysis and Applications》2010,371(2):534-169
Assume that Ω is a bounded domain in RN (N?3) with smooth boundary ∂Ω. In this work, we study existence and uniqueness of blow-up solutions for the problem −Δp(u)+c(x)|∇u|p−1+F(x,u)=0 in Ω, where 2?p. Under some conditions related to the function F, we give a sufficient condition for existence and nonexistence of nonnegative blow-up solutions. We study also the uniqueness of these solutions. 相似文献
12.
Andrea Malchiodi 《Comptes Rendus Mathematique》2004,338(10):775-780
We study positive solutions of the equation ?ε2Δu+u=up, where p>1 and ε>0 is small, with Neumann boundary conditions in a three-dimensional domain . We prove the existence of solutions concentrating along some closed curve on . To cite this article: A. Malchiodi, C. R. Acad. Sci. Paris, Ser. I 338 (2004). 相似文献
13.
We consider the elliptic system Δu=upvq, Δv=urvs in Ω, where p,s>1, q,r>0, and Ω⊂RN is a smooth bounded domain, subject to different types of Dirichlet boundary conditions: (F) u=λ, v=μ, (I) u=v=+∞ and (SF) u=+∞, v=μ on ∂Ω, where λ,μ>0. Under several hypotheses on the parameters p,q,r,s, we show existence and nonexistence of positive solutions, uniqueness and nonuniqueness. We further provide the exact asymptotic behaviour of the solutions and their normal derivatives near ∂Ω. Some more general related problems are also studied. 相似文献
14.
A priori bounds for semilinear equations and a new class of critical exponents for Lipschitz domains
P.J. McKenna 《Journal of Functional Analysis》2007,244(1):220-246
A priori bounds for positive, very weak solutions of semilinear elliptic boundary value problems −Δu=f(x,u) on a bounded domain Ω⊂Rn with u=0 on ∂Ω are studied, where the nonlinearity 0?f(x,s) grows at most like sp. If Ω is a Lipschitz domain we exhibit two exponents p* and p*, which depend on the boundary behavior of the Green function and on the smallest interior opening angle of ∂Ω. We prove that for 1<p<p* all positive very weak solutions are a priori bounded in L∞. For p>p* we construct a nonlinearity f(x,s)=a(x)sp together with a positive very weak solution which does not belong to L∞. Finally we exhibit a class of domains for which p*=p*. For such domains we have found a true critical exponent for very weak solutions. In the case of smooth domains is an exponent which is well known from classical work of Brezis, Turner [H. Brezis, R.E.L. Turner, On a class of superlinear elliptic problems, Comm. Partial Differential Equations 2 (1977) 601-614] and from recent work of Quittner, Souplet [P. Quittner, Ph. Souplet, A priori estimates and existence for elliptic systems via bootstrap in weighted Lebesgue spaces, Arch. Ration. Mech. Anal. 174 (2004) 49-81]. 相似文献
15.
Yoshikazu Kobayashi 《Journal of Mathematical Analysis and Applications》2008,338(2):852-872
A generation theorem of semigroups of locally Lipschitz operators on a subset of a real Banach space is given and applied to the problem of the well-posedness of the Carrier equation utt−κ(‖u‖2)Δu+γ|ut|p−1ut=0 in Ω×(0,∞) with acoustic boundary condition, where p>2 and Ω is a bounded domain in an arbitrary dimensional space. 相似文献
16.
Zhijun Zhang 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(16):5544-5553
We study the existence, boundary behavior and uniqueness of solutions for the singular elliptic system −Δu=u−pv−q,−Δv=u−rv−s,u>0,v>0,x∈Ω,u|∂Ω=v|∂Ω=0, where Ω is a bounded domain with smooth boundary in RN, p,s≥0 and q,r>0. Our results are obtained in a range of p,q,r,s different from those in [M. Ghergu, Lane-Emden systems with negative exponents, J. Funct. Anal. 258 (2010) 3295-3318]. 相似文献
17.
The Cheeger problem for a bounded domain Ω⊂RN, N>1 consists in minimizing the quotients |∂E|/|E| among all smooth subdomains E⊂Ω and the Cheeger constant h(Ω) is the minimum of these quotients. Let be the p-torsion function, that is, the solution of torsional creep problem −Δp?p=1 in Ω, ?p=0 on ∂Ω, where Δpu:=div(|∇u|p−2∇u) is the p-Laplacian operator, p>1. The paper emphasizes the connection between these problems. We prove that . Moreover, we deduce the relation limp→1+‖?p‖L1(Ω)?CNlimp→1+‖?p‖L∞(Ω) where CN is a constant depending only of N and h(Ω), explicitely given in the paper. An eigenfunction u∈BV(Ω)∩L∞(Ω) of the Dirichlet 1-Laplacian is obtained as the strong L1 limit, as p→1+, of a subsequence of the family {?p/‖?p‖L1(Ω)}p>1. Almost all t-level sets Et of u are Cheeger sets and our estimates of u on the Cheeger set |E0| yield |B1|hN(B1)?|E0|hN(Ω), where B1 is the unit ball in RN. For Ω convex we obtain u=|E0|−1χE0. 相似文献
18.
Alexander Gladkov Mohammed Guedda 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(13):4573-4580
In this paper we consider a semilinear parabolic equation ut=Δu−c(x,t)up for (x,t)∈Ω×(0,∞) with nonlinear and nonlocal boundary condition u∣∂Ω×(0,∞)=∫Ωk(x,y,t)uldy and nonnegative initial data where p>0 and l>0. We prove some global existence results. Criteria on this problem which determine whether the solutions blow up in finite time for large or for all nontrivial initial data are also given. 相似文献
19.
Alberto Enciso 《Journal of Mathematical Analysis and Applications》2009,354(2):619-624
Let M be a Riemannian manifold such that its geodesic spheres centered at a point a∈M are isoperimetric and the distance function dist(⋅,a) is isoparametric, and let Ω⊂M be a bounded domain. We prove that if there exists a lower bounded nonconstant function u which is p-harmonic (1<p?n) in the punctured domain Ω?{a} such that both u and are constant on ∂Ω, then u is radial and ∂Ω is a geodesic sphere. The proof hinges on a combination of maximum principles, isoparametricity and the isoperimetric inequality. 相似文献
20.
Goro Akagi 《Journal of Differential Equations》2007,241(2):359-385
The existence of local (in time) solutions of the initial-boundary value problem for the following degenerate parabolic equation: ut(x,t)−Δpu(x,t)−|u|q−2u(x,t)=f(x,t), (x,t)∈Ω×(0,T), where 2?p<q<+∞, Ω is a bounded domain in RN, is given and Δp denotes the so-called p-Laplacian defined by Δpu:=∇⋅(|∇u|p−2∇u), with initial data u0∈Lr(Ω) is proved under r>N(q−p)/p without imposing any smallness on u0 and f. To this end, the above problem is reduced into the Cauchy problem for an evolution equation governed by the difference of two subdifferential operators in a reflexive Banach space, and the theory of subdifferential operators and potential well method are employed to establish energy estimates. Particularly, Lr-estimates of solutions play a crucial role to construct a time-local solution and reveal the dependence of the time interval [0,T0] in which the problem admits a solution. More precisely, T0 depends only on Lr|u0| and f. 相似文献