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Zbl 0671.35023
Zhu, Xiping
Multiple entire solutions of a semilinear elliptic equation. (English)
[J] Nonlinear Anal., Theory Methods Appl. 12, No.11, 1297-1316 (1988). ISSN 0362-546X

By the concentration-compactness principle the author proves the existence of a second entire solution to the semilinear elliptic equation $$-\Delta u+u=q(x)\vert u\vert\sp{\gamma -1}u\quad on\quad {\bbfR}\sp n, $$ $1<\gamma <(n+2)/(n-2)$, if $n\ge 5$, $q(x)\ge q\sb 0\ge 0$, $\lim\sb{x\to \infty} q(x)=q\sb 0$ and $q(x)-q\sb 0\ge c\vert x\vert\sp{- m}$ for $\vert x\vert$ large in addition to the known positive solution given e.g. by {\it P. L. Lions} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1, 109-145 (1984; Zbl 0541.49009) and 223-283 (1984)] or {\it E. S. Noussair} and {\it C. A. Swanson} [Hiroshima Math. J. 15, 127-140 (1985; Zbl 0575.35025)]. Compare also the totally different technique of {\it 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 $-\Delta u=q(x)u\sp{\gamma}$ is proved.
[M.Wiegner]

MSC 2000:: *35J60 Nonlinear elliptic equations
35A05 General existence and uniqueness theorems (PDE)
35J20 Second order elliptic equations, variational methods

Keywords: concentration-compactness principle; existence; entire solution; semilinear; positive solution

Citations: Zbl 0541.49009; Zbl 0575.35025; Zbl 0633.35025

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