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Positive solutions for semipositone (p,N)-Laplacian problems with critical Trudinger–Moser nonlinearities

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Abstract

In this paper, we deal with the existence of positive solutions for semipositone (p,N)-Laplacian problems with critical Trudinger–Moser nonlinearities in a bounded domain:

$$\begin{aligned} \left\{ \begin{array}{clll} -\varDelta _p u-\varDelta _N u=\lambda u^{N-1}e^{\beta u^{N'}} - \mu , &{} \text {in}\,\varOmega ;\\ u>0, &{} \text {in}\,\varOmega ;\\ u=0,&{} \text {on}\,\partial \varOmega . \end{array} \right. \end{aligned}$$

We obtain the positive solutions by combining variational methods with regularity arguments. And the main novelty here is to obtain a uniform \(\mathcal {C}^{1,\alpha }\) priori estimate of the weak solution. Our arguments can be also adapted to seek positive solutions of more general semipositone problems.

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Acknowledgements

We thank the anonymous reviewer for the careful reading of our manuscript and many insightful comments and suggestions. And we thank Kanishka Perera for valuable discussion.

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  1. School of Science, Jiangnan University, Wuxi, Jiangsu, China, 214122

    Yuanyuan Zhang & Yang Yang

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  1. Yuanyuan Zhang
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  2. Yang Yang
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Correspondence to Yang Yang.

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Zhang, Y., Yang, Y. Positive solutions for semipositone (p,N)-Laplacian problems with critical Trudinger–Moser nonlinearities. Rev Mat Complut 35, 133–146 (2022). https://doi.org/10.1007/s13163-021-00386-y

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