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

Li Wang; Jun Wang; Daoguo Zhou; Li Wang; Jun Wang; Daoguo Zhou

doi:10.3934/math.2023690

AIMS Mathematics

2023, Volume 8, Issue 6: 13593-13622. doi: 10.3934/math.2023690

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Research article

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

1.
College of Science, East China Jiaotong University, Nanchang 330013, China
2.
School of Mathematics, Hangzhou Normal University, Hangzhou 311121, China

Received: 16 February 2023 Revised: 20 March 2023 Accepted: 20 March 2023 Published: 10 April 2023
MSC : 35A15, 35B38, 35J60

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.
- double-phase problems,
- penalization method,
- variational methods
Citation: Li Wang, Jun Wang, Daoguo Zhou. Concentration of solutions for double-phase problems with a general nonlinearity[J]. AIMS Mathematics, 2023, 8(6): 13593-13622. doi: 10.3934/math.2023690

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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.

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