Local behaviour of singular solutions for nonlinear elliptic equations in divergence form

Abstract

We prove the uniform boundedness of all solutions for a general class of Dirichlet anisotropic elliptic problems of the formWe assume that f ∈ L r (Ω) with r > N/p. The feature of this study is the inclusion of a possibly singular gradient-dependent term Ψ(u, ∇u) = N j=1 |u| θ j −2 u |∂ju| q j , where θj > 0 and 0 ≤ qj < pj for 1 ≤ j ≤ N . The existence of such weak solutions is contained in a recent paper by the authors.

Singular solutions for divergence-form elliptic equations involving regular variation theory: Existence and classification

“…(ii) λ ∈ (0, ∞). One can show that u has a weak singularity at 0 and can verify (1.16) by using the same argument as in [10, Theorem 5.1] (see also [3,Proposition 6]). We thus omit the details.…”

“…Motivated by previous articles such as [3,9,10,14,27], we aim to obtain a complete understanding of the isolated singularities for nonlinear elliptic equations of the form (1.4) in the punctured unit ball B 1 \{0} in R N (N ≥ 2) under the Assumptions (A 1 )-(A 3 ) given later. A prototype model is A(|x|) = |x| ϑ , b(x) = |x| σ and h(t) = |t| q−1 t for q > p − 1 > 0 and ϑ − σ < p ≤ N + ϑ.…”

Classification of isolated singularities for nonhomogeneous operators in divergence form

“…(1.1) the fact that it is nonhomogeneous asks for a more careful treatment and a different approach involving results coming from the theory of equations analyzed in Orlicz-Sobolev spaces. In particular, we will adapt ideas from Cîrstea and Du [4], Friedman and Véron [6] or Brandolini et al [3] to the new setting.…”

A semilinear singular Sturm–Liouville equation involving measure data

“…For the special case when A(r) = r ϑ with ϑ ∈ (2 − N, 2), a consequence of the main result in [4] is …”

“…After Theorem 1.10 was proved, the author was informed the recent work by Brandolini, Chiacchio, Cîrstea and Trombetti [4]. The authors in [4] studied the positive solutions of the following equation…”