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

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.