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

Abstract

In this paper, we study the following problems with a general nonlinearity: $ \begin{equation*} \label{f} \left\{\begin{aligned} & -\Delta_p u-\Delta_q u+V(\varepsilon x )(|u|^{p-2}u+|u|^{q-2}u) = f(u), &\mathrm{in}\ \mathbb{R}^N, \\ & u\in W^{1, p}( \mathbb{R}^N)\cap W^{1, q}( \mathbb{R}^N), &\mathrm{in}\ \mathbb{R}^N, \end{aligned} \right. \end{equation*} $ where $ \varepsilon > 0 $ is a small parameter, $ 2\leq 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.

Keywords

• double-phase problems
• penalization method
• variational methods