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

Abstract

Let $\Omega \subset {\mathbb{R}}^N$ ($N\ge 1$) be a bounded and smooth domain and $a:\Omega \to \mathbb{R}$ 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 $q\to 1^{-}$. 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 $(P_{a,q})$.

31 pages, 4 figures

References in corpus (1)

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