Positive radial solutions for a noncooperative resonant nuclear reactor model with sign-changing nonlinearities

Abstract

Abstract This paper is concerned with the existence of positive radial solutions of the following resonant elliptic system: { − Δ u = u v + f ( | x | , u ) , 0 < R 1 < | x | < R 2 , x ∈ R N , − Δ v = c g ( u ) − d v , 0 < R 1 < | x | < R 2 , x ∈ R N , ∂ u ∂ n = 0 = ∂ v ∂ n , | x | = R 1 , | x | = R 2 , $$ \textstyle\begin{cases} -\Delta u=uv+f( \vert x \vert ,u), & 0< R_{1}< \vert x \vert < R_{2}, x\in \mathbb{R}^{N}, \\ -\Delta v=cg(u)-dv, & 0< R_{1}< \vert x \vert < R_{2}, x\in \mathbb{R}^{N}, \\ \frac{\partial u}{\partial \textbf{n}}=0= \frac{\partial v}{\partial \textbf{n}},& \vert x \vert =R_{1}, \vert x \vert =R_{2}, \end{cases} $$ where R N $\mathbb{R}^{N}$ ( N ≥ 1 $N\geq 1$ ) is the usual Euclidean space, n indicates the outward unit normal vector, f ∈ C ( [ R 1 , R 2 ] × [ 0 , ∞ ) , R ) $f\in C([R_{1},R_{2}]\times [0,\infty ),\mathbb{R})$ , g ∈ C ( [ 0 , ∞ ) , [ 0 , ∞ ) ) $g\in C([0,\infty ),[0,\infty ))$ , and c and d are positive constants. By employing the classical fixed point theory we establish several novel existence theorems. Our main findings enrich and complement those available in the literature.

Keywords

• Noncooperative models
• Radial solutions
• Resonance
• Existence
• Fixed point