On the improvement of summability of generalized solutions of the Dirichlet problem for nonlinear equations of the fourth order with strengthened ellipticity

Abstract

We consider the Dirichlet problem for a class of nonlinear divergent equations of the fourth order characterized by the condition of strengthened ellipticity imposed on their coefficients. The main result of the present paper shows how the summability of generalized solutions of the given problem improves, depending on the variation in the exponent of summability of the right-hand side of the equation beginning with a certain critical value. The exponent of summability that guarantees the boundedness of solutions is determined more exactly.

Existence of Hölder Continuous Solutions for a Class of Degenerate Fourth-Order Elliptic Equations

“…for almost every x ∈ Ω and for every (η, ξ) ∈ R × R N (2) , by Leray-Lions existence theorem (see [39]), there exists a solution u n ∈ W 1,q 2,p (Ω) of problem (3.1). Moreover, every u n is bounded thanks to the boundedness result of [25] (see also [45]). Now, we are going to prove the following.…”

“…Next, we present a slightly modified version of a well-known Stampacchia's lemma (see [44]), whose proof is contained in [2,25]. See also [21] for new generalizations.…”