Naslov
Singular behaviour of bounded radially symmetric solutions of $p$ - Laplace nonlinear equation

Autori
Pašić, Mervan ; Raguž, Andrija

Izvornik
International Journal of Mathematical Analysis (1334-8671) 3 (2009), 36; 1775-1788

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
radially symmetric solutions ; p-Laplacian ; nonlinear

Sažetak
We study the boundary singular behaviour of radially symmetric solutions $u(x)$ of a class of $p$ - Laplace nonlinear equations: $- \Delta_pu=f(|x|, u, |\nabla u|)$ in a ball $B_R\subset\R^N$, where $u=0$ on $\partial B_R$ and $u\in W^{; ; 1, p}; ; _{; ; loc}; ; (B_R)\cap L^{; ; \infty}; ; (B_R)$. If the nonlinear term $f(x, \eta, \xi)$ satisfies some suitable jumping and singular conditions near $\partial B_R$, we show that box (fractal)-dimension of the graph $G(u)$ of $u(x)$ takes a fractional value $s>N$. It numerically verifies that $G(u)$ is very high concentrated near $\partial B_R$. Next, a kind of singular behavior of $|\nabla u|$ near $\partial B_R$ is established by giving the lower bound for the box-dimension of its graph $G(|\nabla u|)$ which in particular implies $u\notin W^{; ; 1, p}; ; (B_R)$. It generalizes a study on the fractal dimension of the graph of solutions of the one-dimensional $p$ - Laplace nonlinear equation presented in an early paper: Pa\v{; ; s}; ; i\'{; ; c}; ; [J.~Differential Equations~{; ; 190}; ; (2003), 268- 305].

Izvorni jezik
Engleski

Znanstvena područja
Matematika

POVEZANOST RADA