### Existence and exact multiplicity of positive radial solutions of semilinear elliptic problems in annuli

#### Abstract

We study the existence, nonexistence and exact multiplicity of positive radial solutions of the problem: $- \Delta u = f(u)$ in $B(R_1, R)$, where $R_1 > 0$ is a fixed real number and $B(R_1,R) \subset \mathbb R^n (n \geq 3)$, $R > R_1$, is an annulus, for the following boundary conditions: (1) $u = 0$ on $\partial B(R_1,R)$. For $f(t) = t^p - t^q$, $1 < p < q$ and $p \leq (n+2)/(n-2)$, there exist $R_0 \geq \tilde{R} > R_1$ such that the problem admits exactly two solutions for $R > R_0$ and no solution for $R < \tilde{R}$. (2) $u = 0$ on $|x| = R_1$ and $\partial u/ \partial \nu = 0$ on $|x| = R$. (a) For $f(t) = t^p - t^q$, $1 < p < q$ and $p \leq (n+2)/(n-2)$, there exists $R_0 > R_1$ such that the problem admits exactly two solutions for $R > R_0$, exactly one solution for $R = R_0$ and no solution for $R < R_0$. (b) Let $f^{\prime}(t)t > f(t)$. If $f(t) > 0$ for $t > 0$, then the problem admits at most one solution for any $R > R_1$. For a large class of nonlinearities $f$ changing sign (for example $f(t) = t^p -t$), there exists $R_0 > R_1$ such that the problem admits a unique solution for all $R < R_0$. These results have been proved by analyzing the behaviour of the first and the second variations of $u(\gamma,r)$, the unique solution of the corresponding initial value problem with $u^{\prime}(R_1) = \gamma > 0$, with respect to $\gamma$ together with an identity (2.8) which may be regarded as variations of the generalized Pohozaev's identity with respect to $\gamma$.

#### Article information

Source
Adv. Differential Equations, Volume 6, Number 2 (2001), 129-154.

Dates
First available in Project Euclid: 2 January 2013