Existence of a positive solution and numerical solution for some elliptic superlinear problem
In the paper the system \begin{align*} -z^{\Delta\Delta}(t)&= (1+(N-2)\sigma(t))^{\frac{1}{2-N}(2(N-1)+\alpha)}f(z(\sigma(t))),\quad t\in \mathbb{T}_0^+,\\ z(0)&= z^{\Delta}(\infty)=0, \end{align*} is investigated, where $\mathbb{T}_0$ is a time scale so that $0\in \mathbb{T}_0$ and $\sup \mathbb{T}_0=\infty$, $\mathbb{T}_0^+=\mathbb{T}_0\cup [0, \infty)$, $N>2$, $f: \mathbb{R}\to \mathbb{R}^+$ is a continuous function that satisfies the conditions: $f(u)>0$, $u>0$, $\lim_{u\to 0}\frac{f(u)}{u}=0$, $\lim_{u\to\infty} \frac{f(u)}{u}=\infty$, and there exist a rd-continuous function $\phi: \mathbb{T}_0^+\to \mathbb{T}_0^+$, a positive constant $M$ and $\widetilde{\tau}\in \mathbb{T}_0^+$ such that $\int_1^{\infty}\phi(\sigma(\tau))\sigma(\tau)^{-a_N}\Delta\tau<\infty$ and $\frac{f(u\cdot \sigma(\tau))}{f(u)}\leq M \phi(\sigma(\tau))$ for all $u>0$ and for all $\sigma(\tau)>\widetilde{\tau}$, where $a_N= \frac{N+\alpha}{N-2}$. The authors prove that the considered problem has a positive solution. Using the shooting method, numerical simulations of the considered system are presented.
scientific article; zbMATH DE number 6987662
scientific article