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Multiple entire solutions of a semilinear elliptic equation. (English) Zbl 0671.35023

By the concentration-compactness principle the author proves the existence of a second entire solution to the semilinear elliptic equation

-Δu+u=q(x)|u| γ-1 uon n ,

1<γ<(n+2)/(n-2), if n5, q(x)q 0 0, lim x q(x)=q 0 and q(x)-q 0 c|x| -m for |x| large in addition to the known positive solution given e.g. by P. L. Lions [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1, 109-145 (1984; Zbl 0541.49009) and 223-283 (1984)] or E. S. Noussair and C. A. Swanson [Hiroshima Math. J. 15, 127-140 (1985; Zbl 0575.35025)]. Compare also the totally different technique of E. S. Noussair [Bull. Lond. Math. Soc. 19, 443-448 (1987; Zbl 0633.35025)] where the existence of a second positive solution for the slightly different equation -Δu=q(x)u γ is proved.

Reviewer: M.Wiegner
35J60Nonlinear elliptic equations
35A05General existence and uniqueness theorems (PDE) (MSC2000)
35J20Second order elliptic equations, variational methods