On improvement of summability properties of solutions of some degenerate nonlinear fourth-order equations

Abstract

We consider a class of degenerate nonlinear elliptic fourth-order equations in divergence form. Coefficients of the equations satisfy a strengthened ellipticity condition involving two weighted functions. The right-hand side F (x, u) of the equations depends on the unknown function u. Under some conditions, including L^m -summability of F(·, 0) with m close to 1 and restrictions on the exponent σ of the growth of F (x, u) with respect to u, we establish existence of W-solutions of the Dirichlet problem for the given equations and describe the dependence of summability properties of these solutions on m and σ.

Keywords:

• Degenerate nonlinear elliptic fourth-order equations
• Strengthened ellipticity condition
• Dirichlet problem
• Summability of solutions
• AMS Subject Classifications: 35J40
• 35J60
• 35J70