Advances in Differential Equations

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

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