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In this article, we study a class of nonlinear elliptic systems in regular domains of $mathbb{R}^n(ngeq 3)$ with compact boundary. More precisely, we prove the existence of bounded positive continuous solutions to the system $Delta u=lambda f(.,u,v)$, $Delta v=mu g(.,u,v)$, subject to some Dirichlet conditions. Our approach is essentially based on properties of functions in a Kato class $K^{infty }(D)$ and the Schauder fixed point theorem.

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.

Let $D$ be an unbounded domain in $mathbb{R}^{n}$ ($ngeq 2$) with a nonempty compact boundary $partial D$. We consider the following nonlinear elliptic problem, in the sense of distributions, $$displaylines{ Delta u=f(.,u),quad u>0quad hbox{in }D,cr uig|_{partial D}=alpha varphi ,cr lim_{|x|o +infty }frac{u(x)}{h(x)}=eta lambda , }$$ where $alpha ,eta,lambda $ are nonnegative constants with $alpha +eta >0$ and $varphi $ is a nontrivial nonnegative continuous function on $partial D$. The function $f$ is nonnegative and satisfies some appropriate conditions related to a Kato class of functions, and $h$ is a fixed harmonic function in $D$, continuous on $overline{D}$. Our aim is to prove the existence of positive continuous solutions bounded below by a harmonic function. For this aim we use the Schauder fixed-point argument and a potential theory approach.

We present existence results for the polyharmonic nonlinear elliptic boundary-value problem $$displaylines{ (-Delta )^m u=f(cdot,u) quad hbox{in }B cr (frac{partial }{partial u })^j u=0quad hbox{on }partial B, quad 0leq jleq m-1. }$$ (in the sense of distributions), where $B$ is the unit ball in $mathbb{R}^n$ and $ngeq 2$. The nonlinearity $f(x,t)$ satisfies appropriate conditions related to a Kato class of functions $K_{m,n}$. Our approach is based on estimates for the polyharmonic Green function with zero Dirichlet boundary conditions and on the Schauder fixed point theorem.

This paper is concerned with the existence of bounded positive solution for the semilinear elliptic problem Δu=λp(x)f(u) in Ω subject to some Dirichlet conditions, where Ω is a regular domain in ℝn (n≥3) with compact boundary. The nonlinearity f is nonnegative continuous and the potential p belongs to some Kato class K(Ω). So we prove the existence of a positive continuous solution depending on λ by the use of a potential theory approach.

Let D be a bounded domain in ℝn(n≥2). We consider the following nonlinear elliptic problem: Δu=f(⋅,u) in D (in the sense of distributions), u|∂D=ϕ, where ϕ is a nonnegative continuous function on ∂D and f is a nonnegative function satisfying some appropriate conditions related to some Kato class of functions K(D). Our aim is to prove that the above problem has a continuous positive solution bounded below by a fixed harmonic function, which is continuous on D¯. Next, we will be interested in the Dirichlet problem Δu=−ρ(⋅,u) in D (in the sense of distributions), u|∂D=0, where ρ is a nonnegative function satisfying some assumptions detailed below. Our approach is based on the Schauder fixed-point theorem.

We prove the existence of positive explosive solutions for the equation Δu+λ(|x|)|∇u(x)|=ϕ(x,u(x)) in the whole space ℝN(N≥3), where λ:[0,∞)→[0,∞) is a continuous function and ϕ:ℝN×[0,∞)→[0,∞) is required to satisfy some hypotheses detailed below. More precisely, we will give a necessary and sufficient condition for the existence of a positive solution that blows up at infinity.