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Existence and non-existence results for a class of systems under concave-convex nonlinearities
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- Journal:
- Proceedings of the Edinburgh Mathematical Society , First View
- Published online by Cambridge University Press:
- 22 November 2024, pp. 1-29
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In this work, we are interested in studying the following class of problems:(𝒫λμ)
where Ω is a bounded domain in
\begin{align}\left\{\begin{array}{ll}-\Delta u=f_\lambda(x,u,v)& \text{in}~~\Omega\\-\Delta v=g_\mu(x,u,v) & \text{in}~~\Omega\\0\not\equiv u\geq 0,\,\,0\not\equiv v\geq 0& \text{in}~~\Omega\\u=v=0&\text{on}~~\partial\Omega\end{array}\right.\end{align}
$\mathbb{R}^N$, λ > 0, µ > 0, 
$t\mapsto f_\lambda(x,t,t)$ and 
$t\mapsto g_\mu(x,t,t)$ have concave-convex type nonlinearities. We present results related to the existence and non-existence of solutions for problem 
$(\mathcal{P}_{\lambda\mu})$.
