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.

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.

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.

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.

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 $\partial\Omega$.

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.

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.

Abstract:
We are concerned with the multiplicity of solutions of thefollowing singularly perturbed semilinear elliptic equationsin bounded domains Ω:−ε2Δu

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