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

Zhu, Xiping

Nonlinear Anal., Theory Methods Appl. 12, No. 11, 1297-1316 (1988).

By the concentration-compactness principle the author proves the existence of a second entire solution to the semilinear elliptic equation \[ -\Delta u+u=q(x)| u|^{\gamma -1}u\quad on\quad {\mathbb{R}}^ n, \] \(1<\gamma <(n+2)/(n-2)\), if \(n\geq 5\), \(q(x)\geq q_ 0\geq 0\), \(\lim_{x\to \infty} q(x)=q_ 0\) and \(q(x)-q_ 0\geq c| x|^{- m}\) 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 \(-\Delta u=q(x)u^{\gamma}\) is proved.

Reviewer: M.Wiegner

MSC:

35J60	Nonlinear elliptic equations
35A05	General existence and uniqueness theorems (PDE) (MSC2000)
35J20	Variational methods for second-order elliptic equations

Keywords:

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

Citations:

Zbl 0541.49009; Zbl 0575.35025; Zbl 0633.35025

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