# Positive solutions of an elliptic Neumann problem with a sublinear indefinite nonlinearity

## TL;DR

This paper proves the existence of positive solutions for a nonlinear elliptic Neumann problem with a sign-changing weight, analyzes their asymptotic behavior as the nonlinearity approaches linearity, and explores explicit conditions and properties of solutions.

## Contribution

It extends previous existence results for positive solutions of the elliptic Neumann problem with sublinear indefinite nonlinearity and provides detailed asymptotic and explicit solution conditions.

## Key findings

- Existence of positive solutions for q between q_0 and 1.
- Asymptotic behavior of solutions as q approaches 1 from below.
- Conditions ensuring positive solutions in radial symmetric cases.

## Abstract

Let $\Omega\subset\mathbb{R}^{N}$ ($N\geq1$) be a bounded and smooth domain and $a:\Omega\rightarrow\mathbb{R}$ be a sign-changing weight satisfying $\int_{\Omega}a<0$. We prove the existence of a positive solution $u_{q}$ for the problem $(P_{a,q})$: $-\Delta u=a(x)u^{q}$ in $\Omega$, $\frac{\partial u}{\partial\nu}=0$ on $\partial\Omega$, if $q_{0}<q<1$, for some $q_{0}=q_{0}(a)>0$. In doing so, we improve the existence result previously established in [16]. In addition, we provide the asymptotic behavior of $u_{q}$ as $q\rightarrow1^{-}$. When $\Omega$ is a ball and $a$ is radial, we give some explicit conditions on $q$ and $a$ ensuring the existence of a positive solution of $(P_{a,q})$. We also obtain some properties of the set of $q$'s such that $(P_{a,q})$ admits a solution which is positive on $\overline{\Omega}$. Finally, we present some results on nonnegative solutions having dead cores. Our approach combines bifurcation techniques, a priori bounds and the sub-supersolution method. Several methods and results apply as well to the Dirichlet counterpart of $(P_{a,q})$.