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4 can not be global if the initial values are positive somewhere and nonnegative. This completes the solution to the famous Strauss conjecture about semilinear wave equations of the form 
3, and the sub or super critical cases were settled many years earlier by the work of several authors.
doi:10.1016/j.jfa.2006.11.015 
Copyright © 2006 Elsevier Inc. All rights reserved.
A multiplicity theorem for problems with the p-Laplacian
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Received 13 July 2005;
Abstract
We consider a nonlinear elliptic problem driven by the p-Laplacian, with a parameter and a nonlinearity exhibiting a superlinear behavior both at zero and at infinity. We show that if the parameter λ is bigger than λ2=the second eigenvalue of
, then the problem has at least three nontrivial solutions. Our approach combines the method of upper–lower solutions with variational techniques involving the Second Deformation Theorem. The multiplicity result that we prove extends an earlier semilinear (i.e. p=2) result due to Struwe [M. Struwe, Variational Methods, Springer-Verlag, Berlin, 1990].
Keywords: Multiple nontrivial solutions; Superlinear nonlinearity; Upper and lower solutions; Eigenvalues of the p-Laplacian; Second deformation theorem
