A Remark on the Existence of Positive Solution for a Class of (p,q)-Laplacian Nonlinear System with Multiple Parameters and Sign-changing Weight (Journal of Partial Differential Equations, 2013)
The paper deal with the existence of positive solution for the following (p,q)-Laplacian nonlinear system \begin{align*} \left\{  \begin{array}{ll} -Δ_pu=a(x)(α_1f(v)+β_1h(u)), & x∈Ω,\\ -Δ_qv=b(x)(α_2g(u)+β_2k(v)),& x∈Ω,\\ u=v=0,& x∈∂Ω, \end{array} \right. \end{align*} where $Δ_p$ denotes the p-Laplacian operator defined by $Δ_{p}z=div(|∇_z|^{p-2}∇z), p>1, α_1, α_2, β_1, β_2$ are positive parameters and Ω is a bounded domain in $R^N(N > 1)$ with smooth boundary ∂Ω. Here a(x) and b(x) are $C^1$ sign-changing functions that maybe negative near the boundary and f, g, h, k are C^1 nondecreasing functions such that $f, g, h, k: [0,∞)→[0,∞); f (s), g(s), h(s), k(s) > 0; s > 0$ and $lim_{n→∞}\frac{f(Mg(x)^{\frac{1}{q-1}}}{x^{p-1}}=0$ for every $M > 0$.  We discuss the existence of positive solution when $f, g, h, k, a(x)$ and $b(x)$ satisfy certain additional conditions. We use the method of sub-super solutions to establish our results.
Keywords: (p; q)-Laplacian nonlinear system; multiple parameters; sign-changing weight