The Existence and Exact Multiplicity of One-Sign Solutions for Semilinear Elliptic Problems in RNRN

Abstract

In this work, we study the existence of one-sign solutions for the following problem:

$ \left\{\begin{array}{ll}-\Delta u=\lambda a(x) f(u), & \text { in } \mathbb{R}^{N}, \\u(x) \rightarrow 0, & \text { as }|x| \rightarrow+\infty,\end{array}\right.$

where N≥3, λ is a real parameter and $ a \in C_{\mathrm{loc}}^{\alpha}\left(\mathbb{R}^{N}, \mathbb{R}\right)$ for some α∈(0,1) is a weighted function, $ f: \mathbb{R} \rightarrow \mathbb{R}$ is a Hölder continuous function with exponent αα such that f(s)s>0 for any s≠0. 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.

Keywords

Unilateral global bifurcation / one-sign solutions / semilinear elliptic problems / exact multiplicity