Let Ω be a smooth bounded domain in ${𝐑}^n$, n > 1, let a and f be continuous functions on $\overline{\Omega }$, $1^{☆}=\frac{n}{n-1}$. We are concerned here with the existence of solution in $BV\left(\Omega \right)$, positive or not, to the problem:
$$\left\{\begin{array}{cccc}\hfill -\mathrm{div}\sigma +a\left(x\right)signu& amp;={f\left|u\right|}^{1^{☆}-2}u\sigma .\nabla u\hfill & amp;=\left|\nabla u\right|\mathrm{in}\Omega u\mathrm{is}\mathrm{not}\mathrm{identically}\mathrm{zero},& amp;-\sigma .n\left(u\right)=\left|u\right|\mathrm{on}\partial \Omega .\end{array}\right.$$ This problem is closely related to the extremal functions for the problem of the best constant of $W^{1,1}\left(\Omega \right)$ into $L^{\frac{N}{N-1}}\left(\Omega \right)$.

Oscillation theorems are established for forced second order mixed-nonlinear elliptic differential equations ⎧ ${div\left(A\left(x\right)\right|\left|\nabla y\right||}^{p-1}{\nabla y)+⟨b\left(x\right),|\left|\nabla y\right||}^{p-1}\nabla y⟩+C(x,y)=e\left(x\right)$, ⎨ ⎩ $C(x,y)={c\left(x\right)\left|y\right|}^{p-1}y+{\sum }_{i=1}^m{c}_i\left(x\right){\left|y\right|}^{p_i-1}y$$$under quite general conditions. These results are extensions of the recent results of Sun and Wong, [J. Math. Anal. Appl. 334 (2007)] and Zheng, Wang and Han [Appl. Math. Lett. 22 (2009)] for forced second order ordinary differential equations with...