Semilinear Elliptic Differential Equations. Some Recent Results on Global Solutions

Part of the Texts and Monographs in Physics book series (TMP)


All the results on concrete boundary and eigenvalue problems presented in Chaps. 6–8 use in an essential way at least one of the following two assumptions:
  1. 1.

    The domain G ⊂ℝd, over which the problem is considered, is bounded.

  2. 2.
    The exponents which are used to bound the nonlinearities are strictly smaller than the “critical Sobolev exponent” p* defined by
    $$\frac{1}{{p*}} = \frac{1}{p} - \frac{1}{d}{\text{ or }}p* = \frac{{dp}}{{d - p}}$$
    when one looks for solutions in W 1,p(G), 1 ≤ p < d.


Weak Solution Scale Covariance Constrain Minimisation Problem Weak Derivative Critical Sobolev Exponent 
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Further Reading

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Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  1. 1.Theoretische Physik, Fakultät für PhysikUniversität BielefeldBielefeldGermany
  2. 2.Department of MathematicsUniversity of Cape TownRondeboschSouth Africa

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