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On existence of solutions for some classes of elliptic problems with supercritical exponential growth

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Abstract

In the present paper, we study the existence of solutions for the following classes of elliptic problems

where \(\Omega \subset \mathbb {R}^2\) is a smooth bounded domain and

where \(V\in C^0(\mathbb {R}^2)\) is periodic in \(\mathbb {Z}^2\) with \(0\not \in \sigma (-\Delta +V)\). In the both problems above, f is a continuous function of the form

$$\begin{aligned} f(t)=h(t)e^{\alpha _0 |t|^\tau }, \quad t \in \mathbb {R}\end{aligned}$$

for some \(\alpha _0>0\) and \(\tau \ge 2\) and h satisfying some technical conditions. By using variational methods, we show that problems (P) and \((P_V)\) have a nontrivial solution for different types of \(\alpha _0>0\) and \(\tau \ge 2\).

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Acknowledgements

C. O. Alves was supported by CNPq/Brazil 307045/2021-8 and Projeto Universal FAPESQ-PB 3031/2021. L. J. Shen is the corresponding author partially supported by NSFC (12201565). The authors want to thank the two anonymous referees for their deep observations,careful reading and suggestions, which enabled us to improve this version of the manuscript.

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  1. Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, Campina Grande, PB, CEP: 58429-900, Brazil

    Claudianor Oliveira Alves

  2. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, Zhejiang, People’s Republic of China

    Liejun Shen

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Alves, C.O., Shen, L. On existence of solutions for some classes of elliptic problems with supercritical exponential growth. Math. Z. 306, 29 (2024). https://doi.org/10.1007/s00209-023-03420-5

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