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1.
Large solutions of semilinear elliptic equations under the Keller-Osserman condition 总被引:1,自引:0,他引:1
Alan V. Lair 《Journal of Mathematical Analysis and Applications》2007,328(2):1247-1254
We consider the equation Δu=p(x)f(u) where p is a nonnegative nontrivial continuous function and f is continuous and nondecreasing on [0,∞), satisfies f(0)=0, f(s)>0 for s>0 and the Keller-Osserman condition where . We establish conditions on the function p that are necessary and sufficient for the existence of positive solutions, bounded and unbounded, of the given equation. 相似文献
2.
Alan V. Lair 《Journal of Mathematical Analysis and Applications》2011,382(1):324-333
We consider the problem of existence of positive solutions to the elliptic system Δu=p(|x|)vα, Δv=q(|x|)uβ on Rn (n?3) which satisfies . The parameters α and β are positive, and the nonnegative functions p and q are continuous and min{p(r),q(r)} does not have compact support. We show that if αβ?1, then such a solution exists if and only if the functions p and q satisfy
3.
In this paper we find some new conditions to ensure the existence of infinitely many nontrivial solutions for the Dirichlet boundary value problems of the form −Δu+a(x)u=g(x,u) in a bounded smooth domain. Conditions (S1)–(S3) in the present paper are somewhat weaker than the famous Ambrosetti–Rabinowitz-type superquadratic condition. Here, we assume that the primitive of the nonlinearity g is either asymptotically quadratic or superquadratic as |u|→∞. 相似文献
4.
Alan V. Lair 《Journal of Mathematical Analysis and Applications》2008,346(1):99-106
We consider the equation Δu=p(x)uα+q(x)uβ on RN (N?3) where p, q are nonnegative continuous functions and 0<α?β. We establish conditions sufficient to ensure the existence and nonexistence of nonnegative entire large solutions of the equation. 相似文献
5.
This paper is concerned with the structure of the set of radially symmetric solutions for the equation
6.
In this survey article, we recall some known results on existence and multiplicity of sign-changing solutions of elliptic
equations. Methods for obtaining sign-changing solutions developed in the last two decades will also be briefly revisited.
相似文献
7.
This paper deals with the second term asymptotic behavior of large solutions to the problems Δu=b(x)f(u), x∈Ω, subject to the singular boundary condition u(x)=∞, x∈∂Ω, where Ω is a smooth bounded domain in RN, and b(x) is a non-negative weight function. The absorption term f is regularly varying at infinite with index ρ>1 (that is limu→∞f(ξu)/f(u)=ξρ for every ξ>0) and the mapping f(u)/u is increasing on (0,+∞). Our analysis relies on the Karamata regular variation theory. 相似文献
8.
We investigate the existence and the multiplicity of positive solutions for the semilinear elliptic equation −Δu+u=Q(x)|u|p−2u in exterior domain which is very close to RN. The potential Q(x) tends to positive constant at infinity and may change sign. 相似文献
9.
Asymptotic behavior of solutions of semilinear elliptic equations near an isolated singularity 总被引:1,自引:0,他引:1
We consider the semilinear elliptic equation Δu=h(u) in Ω{0}, where Ω is an open subset of (N2) containing the origin and h is locally Lipschitz continuous on [0,∞), positive in (0,∞). We give a complete classification of isolated singularities of positive solutions when h varies regularly at infinity of index q(1,CN) (that is, limu→∞h(λu)/h(u)=λq, for every λ>0), where CN denotes either N/(N−2) if N3 or ∞ if N=2. Our result extends a well-known theorem of Véron for the case h(u)=uq. 相似文献
10.
The existence and multiplicity results are obtained for solutions of a class of the Dirichlet problem for semilinear elliptic equations by the least action principle and the minimax methods, respectively. 相似文献
11.
The following Dirichlet problem (1.1) is considered, where , N≥2, KC2[0,1] and K(r)>0 for 0≤r≤1, , sf(s)>0 for s≠0. Assume moreover that f satisfies the following sublinear condition: f(s)/s>f′(s) for s≠0. A sufficient condition is derived for the uniqueness of radial solutions of (1.1) possessing exactly k−1 nodes, where . It is also shown that there exists KC∞[0,1] such that (1.1) has three radial solutions having exactly one node in the case N=3. 相似文献
12.
Radu Precup 《Journal of Mathematical Analysis and Applications》2011,373(1):138-139
We obtain existence and localization results of positive nontrivial solutions for a class of semilinear elliptic variational systems. The proof is based on variants of Schechter's localized critical point theorems for Hilbert spaces not identified to their duals and on the technique of inverse-positive matrices. The Leray-Schauder boundary condition is also involved. 相似文献
13.
Qiu-yi DAI~ 《中国科学A辑(英文版)》2007,50(8):1141-1156
This paper is devoted to the study of existence,uniqueness and non-degeneracy of positive solutions of semi-linear elliptic equations.A necessary and sufficient condition for the existence of positive solutions to problems is given.We prove that if the uniqueness and non-degeneracy results are valid for positive solutions of a class of semi-linear elliptic equations,then they are still valid when one perturbs the differential operator a little bit.As consequences,some uniqueness results of positive solutions under the domain perturbation are also obtained. 相似文献
14.
Shuibo Huang Qiaoyu Tian Shengzhi Zhang Jinhua Xi Zheng-en Fan 《Nonlinear Analysis: Theory, Methods & Applications》2010,73(11):3489-3501
In this paper, combining the method of lower and upper solutions with the localization method, we establish the boundary blow-up rate of the large positive solutions to the singular boundary value problem
15.
16.
Cheng-yue Li 《高校应用数学学报(英文版)》2009,24(1):49-55
This paper mainly discusses the existence of nontrivial homoclinic solutions for nonperiodic semilinear fourth-order ordinary differential equation
u^(4)+pu″+a(x)u-b(x)u^2=c(x)u^3=3
arising in the study of pattern formation by means of Mountain Pass Lemma. 相似文献
u^(4)+pu″+a(x)u-b(x)u^2=c(x)u^3=3
arising in the study of pattern formation by means of Mountain Pass Lemma. 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.
If g is a nondecreasing nonnegative continuous function we prove that any solution of –u+g(u)=0 in a half plane which blows-up locally on the boundary, in a fairly general way, depends only on the normal variable. We extend this result to problems in the complement of a disk. Our main application concerns the exponential nonlinearity g(u)=eau, or power–like growths of g at infinity. Our method is based upon a combination of the Kelvin transform and moving plane method.Mathematics Subject Classification (1991): 35J60Revised version: 30 June 2004 相似文献
19.
In this paper, we study existence, uniqueness, and asymptotic behavior of large solutions of second-order degenerate elliptic semilinear problems in non-divergence form. The main particularity of the problem is the interior uniform ellipticity of the equation, which degenerates on the boundary, involving an effect on the boundary blow-up profile of the solution. 相似文献
20.
Mohamed Ben Chrouda 《复变函数与椭圆型方程》2018,63(10):1378-1395
We investigate the existence and nonexistence of positive entire large solutions of a class of semilinear equations involving the fractional Laplacian. Sharp lower and upper bounds of the minimal solution, if it exists, are also given. 相似文献