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Existence and uniqueness of solutions of nonlinear elliptic equations without growth conditions at infinity

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Abstract

In this paper, we consider the nonlinear elliptic problem

$$ - \Delta u + {\left| u \right|^{p - 1}}u + {\left| {\nabla u} \right|^q} = f$$

in ℝN, where p > 1 and q > 0. We show that if f ∈ L rloc (ℝN) for suitable r ≥ 1, then there exists a distributional solution of the equation, independently of the behavior of f at infinity. We also analyze the uniqueness of this solution in some cases.

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Authors and Affiliations

  1. Departamento de Matemática, Universidad Técnica Federico Santa María, Casilla V-110, Avda. EspaÑa, 1680, Valparaíso, Chile

    Salomón Alarcón & Jorge García-Melián

  2. Departamento De Análisismatemático, Universidad De Lalaguna, C/. Astrofísico Francisco Sánchez S/N, 38271, La Laguna, Spain

    Alexander Quaas

  3. Instituto Universitario De Estudios Avanzados (Iudea) En Física Atómica Molecular Y Fotónica Facultad De Física Universidad De La Laguna, C/. Astrofísico Francisco Sánchez S/N, 38203, La Laguna, Spain

    Alexander Quaas

Authors
  1. Salomón Alarcón
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  2. Jorge García-Melián
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  3. Alexander Quaas
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Corresponding author

Correspondence to Salomón Alarcón.

Additional information

S. A. was supported by USM Grant # 121002.

A. Q. was partially supported by Fondecyt Grant # 1110210 and CAPDE anillo ACT-125.

All three authors were partially supported by Programa Basal CMM, U. de Chile, and Ministerio de Ciencia e Innovación and FEDER under grant MTM2008-05824 (Spain).

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Alarcón, S., García-Melián, J. & Quaas, A. Existence and uniqueness of solutions of nonlinear elliptic equations without growth conditions at infinity. JAMA 118, 83–104 (2012). https://doi.org/10.1007/s11854-012-0030-6

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  • DOI: https://doi.org/10.1007/s11854-012-0030-6

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