Nonlinear Elliptic Equations in RN without Growth Restrictions on the Data

Abstract

We show existence and regularity of solutions in R N to nonlinear elliptic equations of the form −div A(x, Du) + g(x, u) = f , when f is just a locally integrable function, under appropriate growth conditions on A and g but not on f. Roughly speaking, in the model case −∆ p (u) + |u| s−1 u = f , with p > 2 − (1/N), existence of a nonnegative solution in R N is guaranteed for every nonnegative f ∈ L 1 loc (R N) if and only if s > p − 1.

Existence of solutions of strongly nonlinear elliptic equations in Rn

“…by (3.3)' and since ∇u n p−1,Br ≤ C (lemma 2.2 of [11]). Then, we continue the proof as in lemma 2.2 ([11] part i)) to have the boundedness of (u n ) in W 1,q (B r ).…”

“…In the following, all constants C, C i and C i depends only on the data. We follow the same argument as in lemma 2.1 of [11]. Let K be the N-function defined by K(t) = exp t N 1.…”

“…By combining [10] and [11], we can deduce the existence of weak solutions u for (1.6), with the following regularity: It is our purpose in this paper to prove the limiting regularity W 1,q loc (R N ) and W 1,q 1 loc (R N ) (which is not reached in general ) of weak solutions of (1.6), when we replace the conditions (1.1) and (1.3) by the following:…”

Existence of entropy solutions for degenerate elliptic unilateral problems with variable exponents

“…Otherwise, the term H(x, u, ∇u) is said to be an absorption term, in this case a detailed picture of what happens is available (see e.g. [4,6,8,9,10,11]). …”

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.…”