Existence Results for Strongly Nonlinear Elliptic Equations of Infinite Order

Abstract

In this work, generalized Sobolev spaces and Sobolev spaces of infinite order are considered. Existence of solutions for strongly nonlinear equation of infinite order of the form Au + g(x, u) = f is established. Here A is an elliptic operator from a functional space of Sobolev type to its dual and g(x, s) is a lower order term satisfying a sign condition on s.

Parabolic Equations of Infinite Order withL1Data

“…In view of (28) and (40), we deduce that the family of solutions of problems ( ) is compact in the space (0, , ( , )), where is arbitrary. Consequently, by similar argument as in the elliptic case (using the diagonal process), see [3] or [1], one gets that the sequence converges strongly together with all derivatives to a function ∈ (0, , ∞ 0 ( , )). Letting now > 0 be fixed, a measurable subset of , and > 0, we have 6 Abstract and Applied Analysis In fact, let 0 be a fixed number sufficiently large ( > 0 ) and let V ∈ ∞ (0, ,…”

“…The operator 2 +2 is clearly monotone since the term of higher order of derivation is linear and satisfies the monotonicity condition (see [1,3]). Moreover, thanks to the truncation as in [9] and from assumptions ( 1 ), ( 2 ), and ( 3 ), we deduce that the operator 2 +2 + is bounded, coercive, and pseudo-monotone.…”

“…Next, we use the monotonicity of a part of approximate operator which contains a linear term of higher 2 Abstract and Applied Analysis order of derivation that satisfies the monotonicity condition and prove the existence of solutions in the framework of function space (0, , ∞ 0 ( , )), > 1. Let us mention that an interesting result concerning the stationary counterpart of the problem ( ) has been proved in [3,4].…”

Elliptic equations in weighted Sobolev spaces of infinite order with L 1 data

“…Then, in a similar way as in [9,16], via a diagonalization process, there exists a subsequence, still denoted by u n , which converges uniformly to an element u ∈C ∞ 0 ( ); also for all derivatives D u n → D u holds.…”

Some results on strongly nonlinear anisotropic differential equations

“…in [7], [10] and [14]. Finally, when g does not depend on ∇u, we refer the reader to the recent works [3] and [8], dealing with elliptic equations for a general class of operators of finite and infinite order, and proving the existence of solutions in anisotropic spaces.…”