Nonlinear scalar field equations. II: Existence of infinitely many solutions. (English) Zbl 0556.35046

Berestycki, Henri; Lions, Pierre-Louis

Arch. Ration. Mech. Anal. 82, 347-375 (1983).

[Teil I, ibid., 313-345 (1983; Zbl 0533.35029).] - Es sei \(N\geq 3\) und \(g: {\mathbb{R}}\to {\mathbb{R}}\) sei eine stetige, ungerade Funktion, die den folgenden Bedingungen genügt: \((1)\quad -\infty <\underline{\lim}_{s\to 0}g(s)/s\leq \overline{\lim}_{s\to 0}g(s)/s=- m<0,\) \((2)\quad -\infty \leq \overline{\lim}_{s\to +\infty}g(s)/s^{\ell}\leq 0,\quad \ell =(N+2)/(N-2)\) und \((3)\quad G(\zeta)=\int^{\zeta}_{0}g(s)ds>0\) für ein \(\zeta >0.\)
Betrachtet wird das Problem \[ (*)\quad -\Delta u=g(u)\quad in\quad {\mathbb{R}}^ N,\quad u\in H^ 1({\mathbb{R}}^ N),\quad u\not\equiv 0. \] Als Hauptresultat der Arbeit wird gezeigt, daß dieses Problem eine unendliche Folge von verschiedenen Lösungen \((u_ k)_{k\geq 1}\) mit den folgenden Eigenschaften besitzt: (a) \(u_ k\) ist radialsymmetrisch und gehört zur Klasse \(C^ 2({\mathbb{R}}^ N)\). (b) Es gibt Konstanten \(C_ k\), \(\delta_ k>0\) mit \((4)\quad | D^{\alpha}u_ k(x)| \leq C_ ke^{-\delta_ k| x|},\) \(x\in {\mathbb{R}}^ N\) \((| \alpha | =0,1,2;\quad k\geq 1).\) (c) Es gilt \((5)\quad \lim_{k\to \infty}S(u_ k)=+\infty.\) Dabei ist \((6)\quad S(u)=\int_{{\mathbb{R}}^ N}| \nabla u|^ 2dx-\int_{{\mathbb{R}}^ N}G(u)dx.\) Der Beweis beruht auf einer scharfsinnigen Anwendung funktionalanalytischer Minimaxprinzipien.

Reviewer: E.Heinz

MSC:

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

Keywords:

nonlinear scalar field equations; existence; infinitely many solutions; radial symmetry; estimates; minimax principle

Citations:

Zbl 0533.35029

References:

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