Positive solutions for non-homogeneous semilinear elliptic equations with data that changes sign

2003 ◽  
Vol 133 (2) ◽  
pp. 297-306 ◽  
Author(s):  
Qiuyi Dai ◽  
Yonggeng Gu
Keyword(s):  
Bounded Domain ◽  
Positive Solutions ◽  
Nonlinear Problem ◽  
Elliptic Equations ◽  
Linear Problem ◽  
Semilinear Elliptic Equations ◽  
Semilinear Elliptic ◽  
Existence Of Positive Solutions ◽  
Image Position

Let Ω ⊂ RN be a bounded domain. We consider the nonlinear problem and prove that the existence of positive solutions of the above nonlinear problem is closely related to the existence of non-negative solutions of the following linear problem: .In particular, if p > (N + 2)/(N − 2), then the existence of positive solutions of nonlinear problem is equivalent to the existence of non-negative solutions of the linear problem (for more details, we refer to theorems 1.2 and 1.3 in § 1 of this paper).

Multiple positive solutions of nonhomogeneous semilinear elliptic equations in ℝN

1996 ◽  
Vol 126 (2) ◽  
pp. 443-463 ◽  
Author(s):  
Cao Dao-Min ◽  
Zhou Huan-Song
Keyword(s):  
Positive Solutions ◽  
Elliptic Equations ◽  
Semilinear Elliptic Equations ◽  
Multiple Positive Solutions ◽  
Semilinear Elliptic ◽  
Sobolev Constant ◽  
Image Position

We consider the following problemwhere for all ≦f(x,u)≦c1up-1 + c2u for all x ∈ℝN,u≧0 with c1>0,c2∈(0, 1), 2<p<(2N/(N – 2)) if N ≧ 3, 2 ≧ + ∝ if N = 2. We prove that (*) has at least two positive solutions ifand h≩0 in ℝN, where S is the best Sobolev constant and

scholarly journals On positive solutions of some semilinear elliptic equations

1991 ◽  
Vol 50 (3) ◽  
pp. 343-355 ◽  
Author(s):  
Shin-Hwa Wang ◽  
Nicholas D. Kazarinoff
Keyword(s):  
Lower Bound ◽  
Existence Theorem ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Semilinear Elliptic Equations ◽  
Number Of Solutions ◽  
Nonexistence Theorem ◽  
Semilinear Elliptic ◽  
Existence Of Positive Solutions ◽  
Bound Theorem

AbstractThe existence of positive solutions of some semilinear elliptic equations of the form −Δu = λf(u) is studied. The major results are a nonexistence theorem which gives a λ* = λ*(f,Ω) > 0 below which no positive solutions exist and a lower bound theorem for umax for Ω a ball. As a corollary of the nonexistence theorem that describes the dependence of the number of solutions on λ, two other nonexistence theorems, and an existence theorem are also proved.

A perturbation result of semilinear elliptic equations in exterior strip domains

1997 ◽  
Vol 127 (5) ◽  
pp. 983-1004 ◽  
Author(s):  
Tsing-san Hsu ◽  
Hwai-chiuan Wang
Keyword(s):  
Dirichlet Problem ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Semilinear Elliptic Equations ◽  
Energy Solution ◽  
Perturbation Result ◽  
Semilinear Elliptic ◽  
Image Position

SynopsisIn this paper we show that if the decay of nonzero ƒ is fast enough, then the perturbation Dirichlet problem −Δu + u = up + ƒ(z) in Ω has at least two positive solutions, wherea bounded C1,1 domain S = × ω Rn, D is a bounded C1,1 domain in Rm+n such that D ⊂⊂ S and Ω = S\D. In case ƒ ≡ 0, we assert that there is a positive higher-energy solution providing that D is small.

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