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

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.

### MSC:

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