# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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## Radially symmetric solutions to a superlinear Dirichlet problem in a ball with jumping nonlinearitiesHTML articles powered by AMS MathViewer

by Alfonso Castro and Alexandra Kurepa
Trans. Amer. Math. Soc. 315 (1989), 353-372 Request permission

## Abstract:

Let $p,\varphi :[0,T] \to R$ be bounded functions with $\varphi > 0$. Let $g:{\mathbf {R}} \to {\mathbf {R}}$ be a locally Lipschitzian function satisfying the superlinear jumping condition: (i) ${\lim _{u \to - \infty }}(g(u)/u) \in {\mathbf {R}}$ (ii) ${\lim _{u \to \infty }}(g(u)/{u^{1 + \rho }}) = \infty$ for some $\rho > 0$, and (iii) ${\lim _{u \to \infty }}{(u/g(u))^{N/2}}(NG(\kappa u) - ((N - 2)/2)u \cdot g(u)) = \infty$ for some $\kappa \in (0,1]$ where $G$ is the primitive of $g$. Here we prove that the number of solutions of the boundary value problem $\Delta u + g(u) = p(\left \| x\right \|) + c\varphi (\left \| x\right \|)$ for $x \in {{\mathbf {R}}^N}$ with $\left \| x\right \| < T,u(x) = 0$ for $\left \| x\right \| = T$ tends to $+ \infty$ when $c$ tends to $+ \infty$. The proofs are based on the "energy" and "phase plane" analysis.
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