engleski

Fractal oscillations near the domain boundary of radially symmetric solutions of p-Laplace equations

We consider radially symmetric solutions of $p$-Laplace differential equation $(1)$: $-\Delta_pu=f(|x|)|u|^{; ; p-2}; ; u$ in an annular domain ${; ; \rm \Omega}; ; _{; ; a, b}; ; $. Motivated by [7] and [12], we introduce and study the fractal oscillations near $|x|=b$ of all radially symmetric solutions of equation $(1)$. Precisely, for a given real number $s\in [N, N+1)$ we find some sufficient conditions on the coefficient $f(r)$ such that every radially symmetric nontrivial solution $u(x)$ of equation $(1)$ oscillates near $|x|=b$ and the box-dimension $\dim_B\Gamma (u)$ of the graph $\Gamma(u)$ and corresponding lower and upper $s$-dimensional Minkowski contents ${; ; \mathcal M}; ; _*^s(\Gamma(u))$ and ${; ; \mathcal M}; ; ^{; ; *s}; ; (\Gamma(u))$ satisfy: $\dim_B\Gamma(u)=s$ and $0<{; ; \mathcal M}; ; _*^s(\Gamma(u))\leq {; ; \mathcal M}; ; ^{; ; *s}; ; (\Gamma(u)) < \infty$.

fractal oscillations; Minkowski measurable set; p-Laplacian; radially symmetric function

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