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Semilinear equations in ℝN without condition at infinity

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Abstract

In this paper we establish that some nonlinear elliptic (and parabolic) problems are well posed in all of ℝN without prescribing the behavior at infinity. A typical example is the following: Let 1<p<∞. For everyf ∈ L 1loc (ℝN) there is a uniqueu ∈ L ploc (ℝN) satisfying

$$ - \Delta u + |u|^{p - 1} u = f(x) on \mathbb{R}^N $$

.

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References

  1. Aronson D, Caffarelli L (in press) The initial trace of a solution of the porous medium equation. Trans. Am Math Soc

  2. Baras P, Pierre M (1982) Singularités éliminables d'équations elliptiques semilinéaires, C.R.A.S. Paris 295:519–522; see also Ann Inst Fourier to appear

    Google Scholar 

  3. Bénilan Ph, Brezis H, and Crandall M (1975) A semilinear equation inL 1(ℝN), Ann Sc Norm Sup Pisa 2:523–555

    Google Scholar 

  4. Bénilan Ph, Crandall M, and Pierre M Solutions of the porous medium equation under optimal conditions on initial values. Indiana Univ Math J 33:51–87

  5. Brezis H (1980) Some variational problems of the Thomas-Fermi type. In: Cottle, Giannessi, and Lions (eds) Variational Inequalities and Complementarity Problems. Wiley, New York: 53–73

    Google Scholar 

  6. Brezis H, Strauss W (1973) Semilinear second-order elliptic equations inL 1. J Math Soc Japan 25:565–590

    Google Scholar 

  7. Dahlberg B, Kenig C (in press) Non-negative solutions of the porous medium equation

  8. Gallouet Th, Morel JM (in press) Resolution of a semilinear equation inL 1

  9. Herrero M, Pierre M (in press) Some results for the Cauchy problem foru t = Δu m when 0<m<1

  10. Kato T (1972) Schrödinger operators with singular potentials. Israel J Math 13:135–148

    Google Scholar 

  11. Keller JB (1957) On solutions of Δu = f(u). Comm Pure Appl Math 10:503–510

    Google Scholar 

  12. Loewner C, Nirenberg L (1974) Partial differential equations invariant under conformal or projective transformations. In: Contributions to Analysis. Acad. Press, New York: 245–272

    Google Scholar 

  13. Osserman R (1957) On the inequality Δu ⩾ f(u). Pacific J Math 7:1641–1647

    Google Scholar 

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Communicated by D. Kinderlehrer

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Brezis, H. Semilinear equations in ℝN without condition at infinity. Appl Math Optim 12, 271–282 (1984). https://doi.org/10.1007/BF01449045

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  • DOI: https://doi.org/10.1007/BF01449045

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