Positive solutions for semilinear elliptic equations on expanding annuli: Mountain pass approach. (English) Zbl 0861.35029

Suzuki, Takashi

Funkc. Ekvacioj, Ser. Int. 39, No. 1, 143-164 (1996).

The author studies the elliptic equation \[ -\Delta u=f(u),\quad u>0\quad\text{in }A,\quad u=0\quad\text{on }\partial A,\tag{\(*\)} \] where \(A\) is the annulus \(\{a<|x|<a+ 1\}\subset\mathbb{R}^n\), \(n\geq 4\), \(f\) is a nonlinear function satisfying the following assumptions:
i) \(\lim_{u\to+\infty}(F(u)/u^2)=+\infty\), ii) \((1/2)uf(u)-F(u)\geq \gamma(uf(u)-c^2u^2)\), \((u\geq 0)\), with some \(\gamma>0\), \(0<c<\pi\), where \(F(u)=\int^u_0 f(x)dx\), iii) \(f(u)>0\), \((u\gg 1)\), \(f(0)=0\), \(f'(0)<\pi^2\), iv) \(\lim_{u\to+\infty}(f(u)/u^p)=0\) with some \(p\).
It is proved that infinitely many non-radial solutions arise in \((*)\) as \(a\to+\infty\). To obtain that, the mountain pass lemma is used.

Reviewer: W.Kotarski (Sosnowiec)

MSC:

35J65	Nonlinear boundary value problems for linear elliptic equations
35J20	Variational methods for second-order elliptic equations

Keywords:

infinitely many non-radial solutions; mountain pass lemma