Concentration of solutions for double-phase problems with a general nonlinearity

In this paper, we study the following problems with a general nonlinearity:

$$\{\begin{array}{rlr}& -{\mathrm{\Delta }}_{p}u-{\mathrm{\Delta }}_{q}u+V(\epsilon x)(|u{|}^{p-2}u+|u{|}^{q-2}u)=f(u),& \mathrm{i}\mathrm{n}{\mathbb{R}}^N,\\ & u\in W^{1,p}({\mathbb{R}}^N)\cap W^{1,q}({\mathbb{R}}^N),& \mathrm{i}\mathrm{n}{\mathbb{R}}^N,\end{array}$$

where $\epsilon >0$ is a small parameter, $2\le p<q<N$, the potential $V$ is a positive continuous function having a local minimum. $f:\mathbb{R}\to \mathbb{R}$ is a $C^1$ subcritical nonlinearity. Under some proper assumptions of $V$ and $f,$ we obtain the concentration of positive solutions with the local minimum of $V$ by applying the penalization method for above equation. We must note that the monotonicity of $\frac{f(s)}{s^{p-1}}$ and the so-called Ambrosetti-Rabinowitz condition are not required.