Elliptic problems involving the 1–Laplacian and a singular lower order term

Abstract

This paper is concerned with the Dirichlet problem for an equation involving the 1–Laplacian operator normalΔ1u:=divfalse(Dufalse|Dufalse|false) and having a singular term of the type ffalse(xfalse)uγ. Here f∈LNfalse(normalΩfalse) is nonnegative, 0<γ⩽1 and normalΩ is an open bounded set with Lipschitz‐continuous boundary. We prove an existence result for a concept of solution conveniently defined. The solution is obtained as limit of solutions to p‐Laplacian type problems. Moreover, when f(x)>0 almost everywhere, the solution satisfies those features that might be expected as well as a uniqueness result. We also give explicit one–dimensional examples that show that, in general, uniqueness does not hold. We remark that the Anzellotti theory of L∞–divergence–measure vector fields must be extended to deal with this equation.

The Dirichlet problem for singular elliptic equations with general nonlinearities

“…The non-autonomous case has also been treated; the eigenvalue problem is discussed in [29], the absorption case is handled in [36], while the (sub-)critical exponent problem has been also addressed (see for instance [22,35]). Finally, some early results can be found in [21] concerning the mild singular model case h(s) = s −γ , 0 < γ ≤ 1. As one of our main point also concerns the singular case of a nonlinearity h(s) which is unbounded near the origin s = 0, it is expected that, if f vanishes in a portion of Ω of positive Lebesgue measure, then also the region where u degenerates at zero plays a non-trivial role; in this case, in fact, the characteristic function χ {u>0} may appear in the definition of solution of problem (1.1) (see Definition 6.1 in Section 6 below).…”

Regularizing effect of absorption terms in singular problems

“…We close this section with a lemma which is a slight improvement of a result already contained in [28,29] and which consists in a regularity result for the vector field .…”

“…holds (see [16]). Finally, in [29], the authors prove that if ∈ ∞ loc (Ω) and ∈ BV (Ω) ∩ L ∞ (Ω) such that * ∈ L 1 (Ω div ) then ∈ ∞ (Ω) and a weak trace can be defined as well as a Gauss-Green formula which we recall for the sake of completeness.…”

“…Following the idea in [4], in [45] and [16] the authors prove that ( D ) is well posed if ∈ ∞ (Ω) and ∈ BV (Ω) ∩ L ∞ (Ω) since one can show that * ∈ L ∞ (Ω div ). Moreover in [29] the authors show that (2.1) is well posed if ∈ ∞ loc (Ω) and ∈ BV loc (Ω) ∩ L 1 loc (Ω div ). Moreover, reasoning as in [29], one deduces that, once * ∈ L 1 loc (Ω div ), ( D ) is a Radon measure with local finite total variation satisfying…”

An extension of the pairing theory between divergence-measure fields and BV functions

“…The standard pairing turns out to be a basic tool in many applications. We mention here, among others: extensions of the Gauss-Green formula [6, 8, 9, 13-17, 19, 31]; the setting of the Euler-Lagrange equations associated with integral functionals defined in BV [4,32,33]; Dirichlet problems for equations involving the 1-Laplace operator [5,8,21,22,26,27]; conservation laws [10][11][12][13][14]18]; the Prescribed Mean Curvature problem and capillarity [30,31]; continuum mechanics [9,23,37,38].…”