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Existence and symmetry for elliptic equations in ℝn with arbitrary growth in the gradient

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Abstract

We study the semilinear elliptic equation Δu + g(x, u, Du) = 0 in ℝn. The nonlinearities g can have arbitrary growth in u and Du, including, in particular, exponential behavior. No restriction is imposed on the behavior of g(x, z, p) at infinity except in the variable x. We obtain a solution u which is locally unique and inherits many of the symmetry properties of g. Positivity and asymptotic behavior of the solution are also addressed. Our results can be extended to other domains, such as the half-space and exterior domains. Finally, we give some examples.

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Authors and Affiliations

  1. Universidade Estadual de Campinas, IMECC–Departamento de Matemática, Rua Sérgio Buarque de Holanda, 651, Campinas-SP, CEP 13083-859, Brazil

    Lucas C. F. Ferreira, Marcelo Montenegro & Matheus C. Santos

Authors
  1. Lucas C. F. Ferreira
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  2. Marcelo Montenegro
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  3. Matheus C. Santos
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Correspondence to Lucas C. F. Ferreira.

Additional information

L. C. F. Ferreira, M. Montenegro, and M. C. Santos were partially supported by CNPq/Brazil and Capes/Brazil.

Part of this work was developed while M. Montenegro was visiting IHP and IHÉS under CARMIN program and École Polytechnique, CMLS, France.

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Ferreira, L.C.F., Montenegro, M. & Santos, M.C. Existence and symmetry for elliptic equations in ℝn with arbitrary growth in the gradient. JAMA 130, 1–18 (2016). https://doi.org/10.1007/s11854-016-0027-7

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  • DOI: https://doi.org/10.1007/s11854-016-0027-7

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