Our main purpose is to establish the existence of a positive solution of the system ⎧$-{∆}_{p\left(x\right)}u=F(x,u,v)$, x ∈ Ω, ⎨$-{∆}_{q\left(x\right)}v=H(x,u,v)$, x ∈ Ω, ⎩u = v = 0, x ∈ ∂Ω, where $\Omega \subset {ℝ}^N$ is a bounded domain with C² boundary, $F(x,u,v)={\lambda }^{p\left(x\right)}[g\left(x\right)a\left(u\right)+f\left(v\right)]$, $H(x,u,v)={\lambda }^{q(x})[g\left(x\right)b\left(v\right)+h\left(u\right)]$, λ > 0 is a parameter, p(x),q(x) are functions which satisfy some conditions, and $-{∆}_{p\left(x\right)}u=-{div\left(\right|\nabla u|}^{p\left(x\right)-2}\nabla u)$ is called the p(x)-Laplacian. We give existence results and consider the asymptotic behavior of solutions near the boundary. We do not assume any symmetry conditions on the system.

By a sub-super solution argument, we study the existence of positive solutions for the system ⎧$-{\Delta }_{p}u=a₁\left(x\right)F₁(x,u,v)$ in Ω, ⎪$-{\Delta }_{q}v=a₂\left(x\right)F₂(x,u,v)$ in Ω, ⎨u,v > 0 in Ω, ⎩u = v = 0 on ∂Ω, where Ω is a bounded domain in ${ℝ}^N$ with smooth boundary or $\Omega ={ℝ}^N$. A nonexistence result is obtained for radially symmetric solutions.

We consider the following Kirchhoff type problem involving a critical nonlinearity: ⎧ $-[a+b({\int }_{\Omega }{\left|\nabla u\right|²dx)}^m{]\Delta u=f(x,u)+|u|}^{2*-2}u$ in Ω, ⎨ ⎩ u = 0 on ∂Ω, where $\Omega \subset {ℝ}^N$ (N ≥ 3) is a smooth bounded domain with smooth boundary ∂Ω, a > 0, b ≥ 0, and 0 < m < 2/(N-2). Under appropriate assumptions on f, we show the existence of a positive ground state solution via the variational method.

We discuss the existence of positive radial solutions of the semilinear elliptic equation ⎧-Δu = K(|x|)f(u), x ∈ Ω ⎨αu + β ∂u/∂n = 0, x ∈ ∂Ω, ⎩$li{m}_{\left|x\right|\to \infty }u\left(x\right)=0$, where $\Omega =x\in {ℝ}^N:\left|x\right|>r₀$, N ≥ 3, K: [r₀,∞) → ℝ⁺ is continuous and $0<{\int }_{r₀}^{\infty }rK\left(r\right)dr<\infty$, f ∈ C(ℝ⁺,ℝ⁺), f(0) = 0. Under the conditions related to the asymptotic behaviour of f(u)/u at 0 and infinity, the existence of positive radial solutions is obtained. Our conditions are more precise and weaker than the superlinear or sublinear growth conditions. Our discussion is based on the fixed point...

We study the following singular elliptic equation with critical exponent ⎧$-\Delta u=Q\left(x\right)u^{2*-1}+\lambda u^{-\gamma }$ in Ω, ⎨u > 0 in Ω, ⎩u = 0 on ∂Ω, where $\Omega \subset {ℝ}^N$ (N≥3) is a smooth bounded domain, and λ > 0, γ ∈ (0,1) are real parameters. Under appropriate assumptions on Q, by the constrained minimizer and perturbation methods, we obtain two positive solutions for all λ > 0 small enough.

This paper is devoted to the existence of conformal metrics on $S^3$ with prescribed scalar curvature. We extend well known existence criteria due to Bahri-Coron.