On the solvability of quasilinear elliptic equations of arbitrary order

S. I. Pokhozhaev

Sbornik. Mathematics, Tome 45 (1983) no. 2, pp. 257-271

Résumé

Quasilinear elliptic equations of arbitrary order $2m\geqslant2$ with smooth linear operators under general linear boundary conditions are considered in the space $W_p^{2m}(\Omega)$, $p>1$. Theorems are given presenting a priori estimates of $\|u\|_{2m,p}$ in terms of $\|u\|_{k,\infty}\equiv\sum\limits_{|\gamma|\leqslant k}\sup\limits_\Omega|D^\gamma u(x)|$ with some $k$, $0\leqslant k\leqslant 2m-1$, and in terms of $\|u\|_{m,2}$. For these cases, a critical power law growth is obtained for the nonlinear operator relative to the relevant derivatives. Counterexamples are constructed to show that this critical characteristic growth law cannot be improved without additional assumptions. On the basis of this theory, existence theorems for certain quasilinear elliptic problems are established under the condition that there exists an a priori estimate for $\|u\|_{k,\infty}$ (in the appropriate family of such problems). An existence theorem is also obtained for the solvability of the Dirichlet boundary value problem for some quasilinear elliptic equations of arbitrary order. Examples are given. Bibliography: 11 titles.