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

Abstract

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 $\Omega$ is a bounded smooth domain in ${\mathbb{R}}^N$ ($N\ge 2$), $1<q<2<p$, $\lambda \in \mathbb{R}$, and $a,b\in C^{\alpha }(\overline{\Omega })$ with $0<\alpha <1$. Under some indefinite type conditions on $a$ and $b$ we prove the existence of two nontrivial non-negative solutions for $|\lambda |$ small. We characterize then the asymptotic profiles of these solutions as $\lambda \to 0$, 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 $(P_{\lambda })$.

Cited by in corpus (1)

• Positivity results for indefinite sublinear elliptic problems via a continuity argument