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We prove a finite smooth version of the entropic continuity of Lyapunovexponents of Buzzi-Crovisier-Sarig for $C^\infty$ surface diffeomorphisms [9].As a consequence we show that any $C^r$, $r > 1$, smooth surface diffeomorphism$f$ with $h_{top}(f) > \frac{1}{r} \limsup_n \frac{1}{n} \log^+ \|df^n\|$admits a measure of maximal entropy. We also prove the $C^r$ continuity of thetopological entropy at $f$.

Maximal measure and entropic continuity of Lyapunov exponents for $C^r$ surface diffeomorphisms with large entropy