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An indefinite concave-convex equation under a Neumann boundary condition I

We investigate the problem $$-Δu = λb(x)|u|^{q-2}u +a(x)|u|^{p-2}u \mbox{ in } Ω, \quad \frac{\partial u}{\partial \mathbf{n}} = 0 \mbox{ on } \partial Ω, \leqno{(P_λ)} $$ where $Ω$ is a bounded smooth domain in $\mathbb{R}^N$ ($N \geq2$), $1<q<2<p$, $λ\in \mathbb{R}$, and $a,b \in C^α(\overlineΩ)$ with $0<α<1$. Under some indefinite type conditions on $a$ and $b$ we prove the existence of two nontrivial non-negative solutions for $|λ|$ small. We characterize then the asymptotic profiles of these solutions as $λ\to 0$, which implies in some cases the positivity and ordering of these solutions. In addition, this asymptotic analysis suggests the existence of a loop type subcontinuum in the non-negative solutions set. We prove in some cases the existence of such subcontinuum via a bifurcation and topological analysis of a regularized version of $(P_λ)$.