Abstract: This study is concerned with the following double phase problem $\begin{array}{ll}-{\rm div}(|\nabla u|^{p-2}\nabla u+\mu(x)|\nabla u|^{q-2}\nabla u) +|u|^{p-2}u+\mu(x)|u|^{q-2}u =\lambda f(x,u),\;x\in \mathbb{R} ^N, &\\ \end{array} $ where 1 < p < q < N, $\frac{q}{p}\leq 1+\frac{\alpha}{N}$, $\lambda$ is a real parameter, $0\leq\mu\in C^{0,\alpha}(\mathbb{R} ^N)$ with $\alpha\in(0,1]$ and $f: \mathbb{R} ^N\times\mathbb{R} \rightarrow \mathbb{R} $ satisfies a Carathéodory condition. The aim is to determine the precise positive interval of $\lambda$ for which the problem admits at least one or two nontrivial radially symmetric solutions by applying abstract critical point results.

Key words: Double phase operator, Radially symmetric solutions, Critical point theorems, Variational methods