On some existence theorems for quasilinear elliptic equations

Abstract

Boundary value problems are considered for quasilinear elliptic equations with a principal quasilinear elliptic operator of the second order in the Sobolev space $W_p^2 \left(\Omega\right)$. The main content of the work is to establish the maximum admissible growth of the subordinate nonlinear operator with respect to the corresponding lower-order derivatives. We consider a class of equations for which an a priori estimate of solutions in the norm of the space $\Vert u \Vert_{L_l \left( \Omega\right)}$ $\left( 1 \leq l < \infty \right)$ implies an a priori estimate of the solution in the norm of the space $W_p^2 \left(\Omega\right)$. Based on the theorem on a priori estimates, a general solvability theorem for boundary value problems for quasi-linear elliptic equations is obtained under the condition of the existence of an intermediate a priori estimate $\Vert u \Vert_{L_l \left( \Omega\right)}$ for the solutions of the corresponding family of boundary value problems.