On the Double Phase Variational Problems Without Ambrosetti–Rabinowitz Condition

Abstract

We are concerned with the existence and multiplicity of nontrivial solutions to the following double phase problems:

$$\begin{aligned} \left\{ \begin{array}{ll} -\mathrm{div}(|\nabla u|^{p-2}\nabla u+\alpha (x)|\nabla u|^{q-2}\nabla u)+V(x)|u|^{\gamma -2}u=f(x,u),&{}\ \mathrm{in}\ \Omega ,\\ u=0,&{}\ \mathrm{on}\ {\partial \Omega ,} \end{array}\right. \end{aligned}$$

applying the mountain pass theorem and fountain theorem. The Ambrosetti—Rabinowitz condition as well as the monotonicity of \(f(x,t)/|t|^{q-1}\) are not assumed.

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Acknowledgements

The authors would like to express their sincere thanks to the referees for their valuable comments and suggestions.

Funding

H. B. Chen is supported by the National Natural Science Foundation of China (No. 11671403); J. Yang is Supported by the Research Foundation of Education Bureau of Hunan Province, China (No. 20B457, 19B450) and the National Natural Science Foundation of Hunan Province, China (No. 2019JJ50473).

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The research was carried out in collaboration. All authors read and assured the final manuscript.

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Correspondence to Jie Yang.

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Communicated by Majid Gazor.

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Yang, J., Chen, H. & Liu, S. On the Double Phase Variational Problems Without Ambrosetti–Rabinowitz Condition. Bull. Iran. Math. Soc. (2021). https://doi.org/10.1007/s41980-020-00491-6

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Keywords

  • Double phase problem
  • Ambrosetti–Rabinowitz condition
  • Nontrivial solution

Mathematics Subject Classification

  • 35J20
  • 35J60