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Ground States for the Prescribed Mean Curvature Equation: The Supercritical Case

  • F. V. Atkinson
  • L. A. Peletier
  • J. Serrin
Part of the Mathematical Sciences Research Institute Publications book series (MSRI, volume 12)

Abstract

We shall consider here the question of existence and nonexistence of ground states for the prescribed mean curvature equation in ℝ n (n > 2), that is, we consider solutions of the problem
$$ \left. {\begin{array}{*{20}{c}} {div\left( {\frac{{Du}}{{{{\left( {1 + + {{\left| {Du} \right|}^2}} \right)}^{{1/2}}}}}} \right) + f(u) = 0\;in {\mathbb{R}^n}} \\ {u > 0\quad in\,{\mathbb{R}^n}} \\ {u(x) \to 0\;as\,x \to \infty } \\ \end{array} } \right\} $$
(1.1)
where Du denotes the gradient of u. The function f(u), defined for u > 0, will be assumed throughout to satisfy the following hypotheses:
$$ f \in {C^1}\left[ {0,\infty } \right) $$
(H1)
(H2) f(0) = 0, and there exists a number a ≥ 0 such that
$$ f \in {C^1}\left[ {0,\infty } \right) $$
if a > 0 we require
$$ I(u) = \int\limits_B {F\left( {x,u,Du(x)} \right)} \;dx $$
.

Keywords

Curvature Equation Radial Solution Ground State Solution Nonlinear Diffusion Equation Embed Surface 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc. 1988

Authors and Affiliations

  • F. V. Atkinson
    • 1
    • 2
    • 3
  • L. A. Peletier
    • 1
    • 2
    • 3
  • J. Serrin
    • 1
    • 2
    • 3
  1. 1.Department of MathematicsUniversity of TorontoCanada
  2. 2.Mathematical InstituteUniversity of LeidenThe Netherlands
  3. 3.Department of MathematicsUniversity of MinnesotaUSA

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