Bifurcation and one-sign solutions for semilinear elliptic problems in $ \mathbb{R}^{N} $

Abstract

In this work, we study the existence of one-sign solutions without signum condition for the following problem: $ \begin{eqnarray} \left\{ \begin{array}{ll} -\Delta u = \lambda a(x)f(u), \, \, x\in\mathbb{R}^{N}, & {\rm{}}\ u(x)\rightarrow0, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, {\mathrm{as}}\, \, |x|\rightarrow +\infty, & {\rm{}} \end{array} \right. \end{eqnarray} $ where $ N\geq3 $, $ \lambda $ is a real parameter and $ a\in C^{\alpha}_{loc}(\mathbb{R}^{N}, \mathbb{R}) $ for some $ \alpha\in(0, 1) $ is a weighted function, $ f\in C^{\alpha}(\mathbb{R}, \mathbb{R}) $, and there exist two constants $ s_{2} < 0 < s_{1}, $ such that $ f(s_{1}) = f(s_{2}) = f(0) = 0 $ and $ sf(s) > 0 $ for $ s\in\mathbb{R}\backslash\{s_{1}, 0, s_{2}\}. $ Furthermore, we consider the exact multiplicity of one-sign solutions for above problem under more strict hypotheses. We use bifurcation techniques and the approximation of connected components to prove our main results.

Keywords

• unilateral global bifurcation
• one-sign solutions
• semilinear elliptic problems