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Existence, nonexistence, and asymptotic behavior of solutions for N-Laplacian equations involving critical exponential growth in the whole \({\mathbb {R}}^N\)

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Abstract

In this paper, we are interested in studying the existence or non-existence of solutions for a class of elliptic problems involving the N-Laplacian operator in the whole space. The nonlinearity considered involves critical Trudinger–Moser growth. Our approach is non-variational, and in this way we can address a wide range of problems not yet contained in the literature. Even \(W^{1,N}({\mathbb {R}} ^N)\hookrightarrow L^\infty ({\mathbb {R}} ^N)\) failing, we establish \(\Vert u\Vert _{L^\infty ({\mathbb {R}}^N)} \le C \Vert u\Vert _{W^{1,N}({\mathbb {R}}^N)}^{\Theta }\) (for some \(\Theta >0\)), when u is a solution. To conclude, we explore some asymptotic properties.

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Acknowledgements

The authors are grateful to the referee for his/her valuable suggestions that have improved this article. L.F.O. Faria was partially supported by FAPEMIG CEX APQ 02374/17. A.L.A. de Araujo was partially supported by FAPEMIG FORTIS-10254/2014 and CNPQ.

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  1. Departamento de Matemática, Universidade Federal de Viçosa, Av. Peter Henry Rolfs, s/n, Viçosa, MG, CEP 36570-900, Brazil

    Anderson L. A. de Araujo

  2. Departamento de Matemática, Universidade Federal de Juiz de Fora, ICE, Rua José Lourenço Kelmer, s/n, Juiz de Fora, MG, CEP 36036-900, Brazil

    Luiz F. O. Faria

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Correspondence to Anderson L. A. de Araujo.

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de Araujo, A.L.A., Faria, L.F.O. Existence, nonexistence, and asymptotic behavior of solutions for N-Laplacian equations involving critical exponential growth in the whole \({\mathbb {R}}^N\). Math. Ann. 384, 1469–1507 (2022). https://doi.org/10.1007/s00208-021-02322-3

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