O‘zbekiston matematika jurnali 70-jild 2-son (2026) · 24-32-betlar
On some existence theorems for quasilinear elliptic equations
Amanova, N.
Annotatsiya
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.
interpolation methodSobolev spacequasilinear equation
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