Regularity for solutions of nonlinear elliptic equations with degenerate coercivity

Abstract

We prove regularity results for solutions of some nonlinear Dirichlet problems for an equation in the form divwhere Ω is a bounded open subset of R N , N ≥ 2, α, θ and p are real constants such that: α > 0, 0 ≤ θ ≤ 1 and 1 < p < N . A limit case is also considered.

On L^infty-regularity result for some degenerate nonlinear elliptic equations

“…Observe that if q = +∞, the degeneration of the operator A in problem 2.9 is produced only when the unknown function has large values. Hence, Theorem 4.1 extends the results contained in [2,5,20,21].…”

“…In the non-coercive case where w is a nonzero constant, the existence of bounded weak solutions can be found in [2] with f ∈ L m (Ω), m > max( N p , 1), and g ≡ 0 and in [5] with g ∈ (L r (Ω)) N , r > N p−1…”

Regularizing effects of some lower order terms in non-uniformly nonlinear elliptic equations

“…Due to this lack of coercivity, the classical theory for elliptic operators acting between spaces in duality (see [18]) cannot be applied. However this difficulty is freed in many papers written about Dirichlet problems associated to the noncoercive differential operator A, that it is in the setting of the classic Sobolev spaces (see for instance [1,3,5,6,9,11,12,16,20]) or in the more general setting of Orlicz-Sobolev spaces (see [7,22]). Let us consider the problem div a(x, u, ∇u) = div f in Ω,…”

“…In the case where | f | ∈ L p (Ω), the L (1−θ)N p N − p (Ω)−regularity was proved in [1]. For a solution u of (5), the following regularity results was obtained in [5] in light of various summability of the source term:…”

“…When we replace −div f by data that belong to some Lebesgue spaces, regularity results for solutions of problem (6) was obtained in the coercive case in [2,13] and in [14][15][16] in the noncoercive one.…”