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Transactions of the American Mathematical Society

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

 

 

Radially symmetric solutions to a superlinear Dirichlet problem in a ball with jumping nonlinearities


Authors: Alfonso Castro and Alexandra Kurepa
Journal: Trans. Amer. Math. Soc. 315 (1989), 353-372
MSC: Primary 35J65; Secondary 35B05
DOI: https://doi.org/10.1090/S0002-9947-1989-0933323-8
MathSciNet review: 933323
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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\Vert x\right\Vert) + c\varphi (\left\Vert x\right\Vert)$ for $ x \in {{\mathbf{R}}^N}$ with $ \left\Vert x\right\Vert < T,u(x) = 0$ for $ \left\Vert x\right\Vert = T$ tends to $ + \infty $ when $ c$ tends to $ + \infty $. The proofs are based on the "energy" and "phase plane" analysis.


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DOI: https://doi.org/10.1090/S0002-9947-1989-0933323-8
Keywords: Dirichlet problem, superlinear jumping nonlinearity, singular differential equation, radially symmetric solutions
Article copyright: © Copyright 1989 American Mathematical Society