Existence and L∞-estimates for non-uniformly elliptic equations with non-polynomial growths

Abstract

In the current paper, we investigate the existence and regularity of weak solutions to a class of non-uniformly elliptic equations with degenerate coercivity and non-polynomial growth. The model case is given as follows: 

$$ \mbox{div}\Big( \frac{ \exp(1 + \vert D u\vert )}{(1 + \vert u\vert )^{2}}D u\Big) + \frac{M(\vert D u\vert )}{(1 + \vert u\vert )^{2}}.u=f \quad \mbox{in} \quad \omega.$$An $L^{\infty}$- estimate of solutions is also obtained for an $L^{1}$-datum $f.$