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Existence of positive solutions for semilinear elliptic systems with indefinite weight  [cached]
Ruipeng Chen
Electronic Journal of Differential Equations , 2011,
Abstract: This article concerns the existence of positive solutions of semilinear elliptic system $$displaylines{ -Delta u=lambda a(x)f(v),quadhbox{in }Omega,cr -Delta v=lambda b(x)g(u),quadhbox{in }Omega,cr u=0=v,quad hbox{on } partialOmega, }$$ where $Omegasubseteqmathbb{R}^N (Ngeq1)$ is a bounded domain with a smooth boundary $partialOmega$ and $lambda$ is a positive parameter. $a, b:Omega omathbb{R}$ are sign-changing functions. $f, g:[0,infty) omathbb{R}$ are continuous with $f(0)>0$, $g(0)>0$. By applying Leray-Schauder fixed point theorem, we establish the existence of positive solutions for $lambda$ sufficiently small.
Existence of entire positive solutions for a class of semilinear elliptic systems
Zhijun Zhang
Electronic Journal of Differential Equations , 2010,
Abstract: Under simple conditions on $f_i$ and $g_i$, we show the existence of entire positive radial solutions for the semilinear elliptic system $$displaylines{ Delta u =p(|x|)f_1(v)f_2(u)cr Delta v =q(|x|)g_1(v)g_2(u), }$$ where $xin mathbb{R}^N$, $Ngeq 3$, and p,q are continuous functions.
Existence and uniqueness conditions of positive solutions to semilinear elliptic equations with double power nonlinearities  [PDF]
Shinji Kawano
Mathematics , 2008,
Abstract: In this article we find the equivalent conditions to assure the existence and uniqueness of positive solutions to semilinear elliptic equations wih double power nonlinearities. As a bonus, we give a simpler proof of our former result that the uniqueness condition comes from the existence condition.
Existence and asymptotic behaviour of positive solutions for semilinear elliptic systems in the Euclidean plane
Abdeljabbar Ghanmi,Faten Toumi
Electronic Journal of Differential Equations , 2011,
Abstract: We study the semilinear elliptic system $$ Delta u=lambda p(x)f(v),Delta v=lambda q(x)g(u), $$ in an unbounded domain D in $ mathbb{R}^2$ with compact boundary subject to some Dirichlet conditions. We give existence results according to the monotonicity of the nonnegative continuous functions f and g. The potentials p and q are nonnegative and required to satisfy some hypotheses related on a Kato class.
Existence of strictly positive solutions for sublinear elliptic problems in bounded domains  [PDF]
Tomas Godoy,Uriel Kaufmann
Mathematics , 2013,
Abstract: Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^{N}$ and let $m$ be a possibly discontinuous and unbounded function that changes sign in $\Omega$. Let $f:\left[ 0,\infty\right) \rightarrow\left[ 0,\infty\right) $ be a continuous function such that $k_{1}\xi^{p}\leq f\left(\xi\right) \leq k_{2}\xi^{p}$ for all $\xi\geq0$ and some $k_{1},k_{2}>0$ and $p\in\left(0,1\right) $. We study existence and nonexistence of strictly positive solutions for nonlinear elliptic problems of the form $-\Delta u=m\left(x\right) f\left(u\right) $ in $\Omega$, $u=0$ on $\partial\Omega$.
Existence and uniqueness of solutions to a semilinear elliptic system
Zu-Chi Chen,Ying Cui
Electronic Journal of Differential Equations , 2009,
Abstract: In this article, we show the existence and uniqueness of smooth solutions for boundary-value problems of semilinear elliptic systems.
Existence of positive bounded solutions for some nonlinear polyharmonic elliptic systems  [cached]
Sabrine Gontara,Zagharide Zine El Abidine
Electronic Journal of Differential Equations , 2010,
Abstract: We prove existence results for positive bounded continuous solutions of a nonlinear polyharmonic system by using a potential theory approach and properties of a large functional class $K_{m,n}$ called Kato class.
Infinitely many solutions for semilinear nonlocal elliptic equations under noncompact settings  [PDF]
Woocheol Choi,Jinmyoung Seok
Mathematics , 2014,
Abstract: In this paper, we study a class of semilinear nonlocal elliptic equations posed on settings without compact Sobolev embedding. More precisely, we prove the existence of infinitely many solutions to the fractional Brezis-Nirenberg problems on bounded domain.
Multiple solutions for singularly perturbed semilinear elliptic equations in bounded domains  [PDF]
Michinori Ishiwata
Abstract and Applied Analysis , 2005, DOI: 10.1155/aaa.2005.185
Abstract: We are concerned with the multiplicity of solutions of thefollowing singularly perturbed semilinear elliptic equationsin bounded domains Ω:−ε2Δu
Multiple Positive Solutions for Semilinear Elliptic Equations in Involving Concave-Convex Nonlinearities and Sign-Changing Weight Functions  [PDF]
Tsing-San Hsu,Huei-Li Lin
Abstract and Applied Analysis , 2010, DOI: 10.1155/2010/658397
Abstract: We study the existence and multiplicity of positive solutions for the following semilinear elliptic equation in , , where , if , if ), , satisfy suitable conditions, and may change sign in . 1. Introduction and Main Results In this paper, we deal with the existence and multiplicity of positive solutions for the following semilinear elliptic equation: where , ( if , if ), and are measurable functions and satisfy the following conditions: with in ; and in . Semilinear elliptic equations with concave-convex nonlinearities in bounded domains are widely studied. For example, Ambrosetti et al. [1] considered the following equation: where , . They proved that there exists such that ( ) admits at least two positive solutions for all and has one positive solution for and no positive solution for . Actually, Adimurthi et al. [2], Damascelli et al. [3], Ouyang and Shi [4], and Tang [5] proved that there exists such that ( ) in the unit ball has exactly two positive solutions for and has exactly one positive solution for and no positive solution exists for . For more general results of ( ) (involving sign-changing weights) in bounded domains see Ambrosetti et al. [6], García Azorero et al. [7], Brown and Wu [8], Brown and Zhang [9], Cao and Zhong [10], de Figueiredo et al. [11], and their references. However, little has been done for this type of problem in . We are only aware of the works [12–16] which studied the existence of solutions for some related concave-convex elliptic problems (not involving sign-changing weights). Furthermore, we do not know of any results for concave-convex elliptic problems involving sign-changing weight functions except [17]. Wu in [17] has studied the multiplicity of positive solutions for the following equation involving sign-changing weights: where , the parameters . He also assumed that is sign-chaning and , where and satisfy suitable conditions, and proved that ( ) has at least four positive solutions. The main aim of this paper is to study ( ) in involving concave-convex nonlinearities and sign-changing weight functions. We will discuss the Nehari manifold and examine carefully connection between the Nehari manifold and the fibrering maps; then using arguments similar to those used in [18], we will prove the existence of two positive solutions by using Ekeland’s variational principle [19]. Set where , , and is the best Sobolev constant for the imbedding of into . Now, we state the first main result about the existence of positive solution of ( ) in . Theorem 1.1. Assume that (A1) and (B1) hold. If , then ( ) admits at least one
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