Existence of positive radial solutions of general quasilinear elliptic systems - Devine Daniel (Advances in Nonlinear Analysis, 2025)
Let Ω⊂Rn(n≥2)\Omega \subset {{\mathbb{R}}}^{n}\hspace{0.33em}\left(n\ge 2) be either an open ball BR{B}_{R} centred at the origin or the whole space. We study the existence of positive, radial solutions of quasilinear elliptic systems of the form Δpu=f1(∣x∣)g1(v)∣∇u∣αinΩ,Δpv=f2(∣x∣)g2(v)h(∣∇u∣)inΩ,\left\{\begin{array}{rcl}{\Delta }_{p}u& =& {f}_{1}\left(| x| ){g}_{1}\left(v){| \nabla u| }^{\alpha }\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ {\Delta }_{p}v& =& {f}_{2}\left(| x| ){g}_{2}\left(v)h\left(| \nabla u| )\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\end{array}\right. where α≥0\alpha \ge 0, Δp{\Delta }_{p} is the pp-Laplace operator, p>1p\gt 1, and for i,j=1i,j=1, 2, we assume fi,gj{f}_{i},{g}_{j}, and hh are continuous, non-negative and non-decreasing functions. For functions gj{g}_{j} which grow polynomially, we prove sharp conditions for the existence of positive radial solutions which blow up at ∂BR\partial {B}_{R}, and for the existence of global solutions.