A multiplicity result for double phase problem in the whole space

In the present paper, we discuss the solutions of the following double phase problem

$$-\mathrm{d}\mathrm{i}\mathrm{v}(|\mathrm{\nabla }u{|}^{{}^{p-2}}\mathrm{\nabla }u+\mu (x)|\mathrm{\nabla }u{|}^{{}^{q-2}}\mathrm{\nabla }u)+|u{|}^{{}^{p-2}}u+\mu (x)|u{|}^{{}^{q-2}}u=f(x,u),x\in {\mathbb{R}}^N,$$

where $N\ge 2$, $1<p<q<N$ and $0\le \mu \in C^{{}^{0,\alpha }}({\mathbb{R}}^N),\alpha \in (0,1]$. Based on the theory of the double phase Sobolev spaces $W^{{}^{1,H}}({\mathbb{R}}^N)$, we prove the existence of at least two non-trivial weak solutions.