The Existence and Exact Multiplicity of One-Sign Solutions for Semilinear Elliptic Problems in $\mathbb{R}^N$

Abstract

In this work, we study the existence of one-sign solutions for the following problem: $$ \begin{cases} -\Delta u = \lambda a(x) f(u), & \text{in } \mathbb{R}^N, \\ u(x) \to 0, & \text{as } |x| \to +\infty. \end{cases} $$where $N≥3$, $λ$ is a real parameter and $a∈C_{loc}^α$ ($\mathbb{R}^N$,$\mathbb{R}$) for some $α∈(0,1)$ is a weighted function, $f: \mathbb{R}$→$\mathbb{R}$ is a Hölder continuous function with exponent $α$ such that $f(s)s$>0 for any $s\ne0$. We determine the intervals of $λ$ for the existence, exact multiplicity of one-sign solutions for this problem. We use bifurcation techniques and the approximation of connected components to prove our main results.