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Existence and exact multiplicity of positive radial solutions of semilinear elliptic problems in annuli. (English) Zbl 1064.35060
The author studies the existence, nonexistence and exact multiplicity of positive radial solutions of the equation $-\Delta u = f(u) \quad\text{in}\quad B(R_1,R)=\{x\in{\mathbb R}^n: R_1<| x| <R \} (*)$ where $$n\geq 3$$, $$R_1>0$$ is fixed, and $$f(t)=t^p-t^q$$ where $$1<p<q$$ and $$p\leq (n+2)/(n-2)$$. The two main results are the following.
(i) For the boundary condition $$u=0$$ on $$\partial B(R_1,R)$$ there exist $$R_0\geq \tilde R > R_1$$ such that $$(*)$$ has exactly two positive radial solutions if $$R>R_0$$ and no such solution if $$R<\tilde R$$.
(ii) For the boundary condition $$u=0$$ on $$| x| =R_1$$ and $$\partial u/\partial\nu=0$$ on $$| x| =R$$ there exists $$R_0 > R_1$$ such that $$(*)$$ has exactly two positive radial solutions if $$R>R_0$$, exactly one such solution if $$R=R_0$$ and no solution if $$R<R_0$$.
##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35J60 Nonlinear elliptic equations 35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs