Abstract
We consider the following double phase problem:
where \(N\ge 2,\) and \(\frac{q}{p}<1+\frac{1}{N}, a:\mathbb {R}^{N}\rightarrow [0,+\infty )~\mathrm {is~Lipschitz~continuous.}\) The Ambrosetti–Rabinowitz type condition, that is so-called (AR) condition: there exist \(L>0,\theta >q\) such that for \(|t|\ge L\) and a.e. \(x\in \mathbb {R}^{N}\),
as well as the monotonicity of \(f(x,t)/|t|^{q-1}\) are not assumed. Under appropriate assumptions on V and f, we prove that the above problem has at least a nontrivial solution.
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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 of China (Nos. 17C1263 and 19B450) and the National Natural Science Foundation of Hunan Province of China (No. 2019JJ50473).
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Yang, J., Chen, H. & Liu, S. Existence of solutions for the double phase variational problems without AR-condition. manuscripta math. 165, 505–519 (2021). https://doi.org/10.1007/s00229-020-01228-9
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DOI: https://doi.org/10.1007/s00229-020-01228-9