共查询到20条相似文献,搜索用时 23 毫秒
1.
Soohyun Bae 《Journal of Differential Equations》2009,247(5):1616-1635
We establish that the elliptic equation Δu+K(x)up+μf(x)=0 in Rn has a continuum of positive entire solutions for small μ?0 under suitable conditions on K, p and f. In particular, K behaves like l|x| at ∞ for some l?−2, but may change sign in a compact region. For given l>−2, there is a critical exponent pc=pc(n,l)>1 in the sense that the result holds for p?pc and involves partial separation of entire solutions. The partial separation means that the set of entire solutions possesses a non-trivial subset in which any two solutions do not intersect. The observation is well known when K is non-negative. The point of the paper is to remove the sign condition on compact region. When l=−2, the result holds for any p>1 while pc is decreasing to 1 as l decreases to −2. 相似文献
2.
Zhijun Zhang 《Journal of Mathematical Analysis and Applications》2005,308(2):532-540
By constructing the comparison functions and the perturbed method, it is showed that any solution u∈C2(Ω) to the semilinear elliptic problems Δu=k(x)g(u), x∈Ω, u|∂Ω=+∞ satisfies , where Ω is a bounded domain with smooth boundary in RN; , −2<σ, c0>0, ; g∈C1[0,∞), g?0 and is increasing on (0,∞), there exists ρ>0 such that , ∀ξ>0, , . 相似文献
3.
In this article, we discuss the blow-up problem of entire solutions of a class of second-order quasilinear elliptic equation Δ p u ≡ div(|?u| p?2?u) = ρ(x)f(u), x ∈ R N . No monotonicity condition is assumed upon f(u). Our method used to get the existence of the solution is based on sub-and supersolutions techniques. 相似文献
4.
5.
In this paper, we use new analyses to assert that there are three positive solutions of Eq. (1.1) in infinite cylinder domain with hole . 相似文献
6.
Haitao Yang 《Journal of Mathematical Analysis and Applications》2006,314(1):85-96
In this paper we consider the semilinear elliptic problem Δu=a(x)f(u), u?0 in Ω, with the boundary blow-up condition u|∂Ω=+∞, where Ω is a bounded domain in RN(N?2), a(x)∈C(Ω) may blow up on ∂Ω and f is assumed to satisfy (f1) and (f2) below which include the sublinear case f(u)=um, m∈(0,1). For the radial case that Ω=B (the unit ball) and a(x) is radial, we show that a solution exists if and only if . For Ω a general domain, we obtain an optimal nonexistence result. The existence for nonradial solutions is also studied by using sub-supersolution method. 相似文献
7.
8.
Soohyun Bae 《Journal of Differential Equations》2007,237(1):159-197
We establish that for n?3 and p>1, the elliptic equation Δu+K(x)up=0 in Rn possesses a continuum of positive entire solutions with logarithmic decay at ∞, provided that a locally Hölder continuous function K?0 in Rn?{0}, satisfies K(x)=O(σ|x|) at x=0 for some σ>−2, and 2|x|K(x)=c+O([log|x|]−θ) near ∞ for some constants c>0 and θ>1. The continuum contains at least countably many solutions among which any two do not intersect. This is an affirmative answer to an open question raised in [S. Bae, T.K. Chang, On a class of semilinear elliptic equations in Rn, J. Differential Equations 185 (2002) 225-250]. The crucial observation is that in the radial case of K(r)=K(|x|), two fundamental weights, and , appear in analyzing the asymptotic behavior of solutions. 相似文献
9.
Jesse D. Peterson Aihua W. Wood 《Journal of Mathematical Analysis and Applications》2011,384(2):284-292
We present the existence of entire large positive radial solutions for the non-monotonic system Δu=p(|x|)g(v), Δv=q(|x|)f(u) on Rn where n?3. The functions f and g satisfy a Keller-Osserman type condition while nonnegative functions p and q are required to satisfy the decay conditions and . Further, p and q are such that min(p,q) does not have compact support. 相似文献
10.
11.
Hong Li 《Journal of Mathematical Analysis and Applications》2010,365(1):338-333
Under the simple conditions on f and g, we show that entire positive radial solutions exist for the semilinear elliptic system Δu=p(|x|)f(v), Δv=q(|x|)g(u), x∈RN, N?3, where the functions are continuous. 相似文献
12.
Existence and uniqueness of large positive solutions are obtained for some semilinear elliptic equations with critical and supercritical growth on general bounded smooth domains. It is shown that the large positive solution develops a boundary layer. The boundary derivative estimate of the large solution is also established. 相似文献
13.
14.
We consider a special class of radial solutions of semilinear equations −Δu=g(u) in the unit ball of Rn. It is the class of semi-stable solutions, which includes local minimizers, minimal solutions, and extremal solutions. We establish sharp pointwise, Lq, and Wk,q estimates for semi-stable radial solutions. Our regularity results do not depend on the specific nonlinearity g. Among other results, we prove that every semi-stable radial weak solution is bounded if n?9 (for every g), and belongs to H3=W3,2 in all dimensions n (for every g increasing and convex). The optimal regularity results are strongly related to an explicit exponent which is larger than the critical Sobolev exponent. 相似文献
15.
For n≥3 and p>1, the elliptic equation Δu+K(x)up+μf(x)=0 in possesses a continuum of positive entire solutions, provided that (i) locally Hölder continuous functions K and f vanish rapidly, for instance, K(x),f(x)=O(|x|l) near ∞ for some l<−2 and (ii) μ≥0 is sufficiently small. Especially, in the radial case with K(x)=k(|x|) and f(x)=g(|x|) for some appropriate functions k,g on [0,∞), there exist two intervals Iμ,1, Iμ,2 such that for each α∈Iμ,1 the equation has a positive entire solution uα with uα(0)=α which converges to l∈Iμ,2 at ∞, and uα1<uα2 for any α1<α2 in Iμ,1. Moreover, the map α to l is one-to-one and onto from Iμ,1 to Iμ,2. If K≥0, each solution regarded as a steady state for the corresponding parabolic equation is stable in the uniform norm; moreover, in the radial case the solutions are also weakly asymptotically stable in the weighted uniform norm with weight function |x|n−2. 相似文献
16.
In this paper, we study the effect of domain shape on the number of positive and nodal (sign-changing) solutions for a class of semilinear elliptic equations. We prove a semilinear elliptic equation in a domain Ω that contains m disjoint large enough balls has m2 2-nodal solutions and m positive solutions. 相似文献
17.
Jiqin Deng 《Journal of Mathematical Analysis and Applications》2007,336(2):1395-1405
In this paper, by using the fixed point theory, under quite general conditions on the nonlinear term, we obtain an existence result of bounded positive solutions of semilinear elliptic equations in exterior domain of Rn, n?3. 相似文献
18.
Qing Miao 《Applicable analysis》2013,92(12):1893-1905
For a given bounded domain Ω in R N with smooth boundary ?Ω, we give sufficient conditions on f so that the m-Laplacian equation △ m u = f(x, u, ?u) admits a boundary blow-up solution u ∈ W 1,p (Ω). Our main results are new and extend the results in J.V. Concalves and Angelo Roncalli [Boundary blow-up solutions for a class of elliptic equations on a bounded domain, Appl. Math. Comput. 182 (2006), pp. 13–23]. Our approach employs the method of lower–upper solution theorem, fixed point theory and weak comparison principle. 相似文献
19.
We consider the semilinear elliptic equation Δu+K(|x|)up=0 in RN for N>2 and p>1, and study separation phenomena of positive radial solutions. With respect to intersection and separation, we establish a classification of the solution structures, and investigate the structures of intersection, partial separation and separation. As a consequence, we obtain the existence of positive solutions with slow decay when the oscillation of the function r−?K(r) with ?>−2 around a positive constant is small near r=∞ and p is sufficiently large. Moreover, if the assumptions hold in the whole space, the equation has the structure of separation and possesses a singular solution as the upper limit of regular solutions. We also reveal that the equation changes its nature drastically across a critical exponent pc which is determined by N and the order of the behavior of K(r) as r=|x|→0 and ∞. In order to understand how subtle the structure is on K at p=pc, we explain the criticality in a similar way as done by Ding and Ni (1985) [6] for the critical Sobolev exponent p=(N+2)/(N−2). 相似文献
20.
Tsing-San Hsu 《Journal of Mathematical Analysis and Applications》2007,332(2):814-832
In this paper, assume that h is nonnegative and ‖hL2‖>0, we prove that if ‖hL2‖ is sufficiently small, then there are at least three positive solutions of Eq. (1) in an exterior cylinder domain. 相似文献