Finite and infinite energy solutions of singular elliptic problems: Existence and uniqueness

Abstract

We establish existence and uniqueness of solution for the homogeneous Dirichlet problem associated to a fairly general class of elliptic equations modeled bywhere f is an irregular datum, possibly a measure, and h is a continuous function that may blow up at zero. We also provide regularity results on both the solution and the lower order term depending on the regularity of the data, and we discuss their optimality.

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

“…where L denotes the Lipschitz constant of h. Moreover, it follows from Remark 4.8. The BV -estimate proven in Step 2 depends only on the fact that γ > 0 (see (37)). We point out that in the nonsingular setting (when γ = 0) this fact does not hold, unless f is small enough.…”

“…Problem (2) for p = 2 was initially proposed in 1960 in the pioneering work by Fulks and Maybee [21] as a model for several physical situations. This problem was then studied by many authors, among which we will quote the works of Stuart [42], Crandall, Rabinowitz and Tartar [16], Lazer and McKenna [30], Boccardo and Orsina [8], Coclite and Coclite [15], Arcoya and Moreno-Mérida [6], Oliva and Petitta [37], and Giachetti, Martinez-Aparicio and Murat [22][23][24][25].…”

Regularizing effect of absorption terms in singular problems

“…Otherwise if is not sufficiently regular we are just able to prove that, in general, every truncation of the solution has locally finite energy. For this kind of effects and even more we refer to [11,26,47,48]. When = 1 the same effect arises: for instance in [28] it is proved the existence of a locally BV -solution if is in L N (Ω).…”

“…Moreover if γ = 1 the solution is always in H 1 0 (Ω) even if is just an L 1 -function as one can formally deduce by taking itself as test function in (1.1) while if γ > 1 the solution belongs only locally to H 1 (Ω) and the boundary datum is given as a suitable power of the solution having zero Sobolev trace. Then, in [48], a general (as above) is considered and various results are proved depending on γ, θ, and in order to have finite energy solutions. Among other things, one of the results concerning finite energy solutions can be summarized as follows: let 0 ≤ γ ≤ 1 then ∈ H 1 0 (Ω) if 0 ≤ θ < 1 and ∈ L ( 2 * 1−θ ) ′ (Ω) or if θ ≥ 1 and is just an L 1 -function.…”

A nonlinear parabolic problem with singular terms and nonregular data

“…Using dominated convergence theorem one can pass to the limit also on the right hand side of (4.2) and we conclude. (Ω) (see [22,6,31]), one has, by Lemma 4.2, that ( ) = ( ) turns out to be a finite energy solution of (P 0 ).…”