A curve of positive solutions for an indefinite sublinear Dirichlet problem

TL;DR

This paper studies the existence and behavior of positive solutions for an indefinite sublinear Dirichlet problem, revealing a solution curve and asymptotic properties as the parameter varies, using bifurcation and sub-supersolutions methods.

Contribution

It introduces a solution curve for the problem with sign-changing weight and analyzes its asymptotic behavior, also connecting to ground state solutions and singular problems.

Findings

Existence of a continuous curve of positive solutions for q in (0,1)

Asymptotic behavior of solutions as q approaches 0 and 1

Conditions under which solutions are ground states

Abstract

We investigate the existence of a curve $q\mapsto u_{q}$, with $q\in(0,1)$, of positive solutions for the problem $(P_{a,q})$: $-\Delta u=a(x)u^{q}$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a bounded and smooth domain of $\mathbb{R}^{N}$ and $a:\Omega\rightarrow\mathbb{R}$ is a sign-changing function (in which case the strong maximum principle does not hold). In addition, we analyze the asymptotic behavior of $u_{q}$ as $q\rightarrow0^{+}$ and $q\rightarrow1^{-}$. We also show that in some cases $u_{q}$ is the ground state solution of $(P_{a,q})$. As a byproduct, we obtain existence results for a singular and indefinite Dirichlet problem. Our results are mainly based on bifurcation and sub-supersolutions methods.