Elliptic equations with general singular lower order term and measure data

We study existence and uniqueness of solutions to a nonlinear elliptic boundary value problem with a general, and possibly singular, lower order term, whose model isHere Ω is an open bounded subset of R N (N ≥ 2), ∆ p u := div(|∇u| p−2 ∇u) (1 < p < N ) is the p-laplacian operator, µ is a nonnegative bounded Radon measure on Ω and H(s) is a continuous, positive and finite function outside the origin which grows at most as s −γ , with γ ≥ 0, near zero.2010 Mathematics Subject Classification. 35J60, 35J61, 35J75, 35R06.

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“…Proof. The proof is similar to the one of Lemma 3.4 in [22] given for p = 2. For this reason we just sketch it.…”

Section: The Approximation Scheme If H Is Singularmentioning

confidence: 68%

Existence and uniqueness results for possibly singular nonlinear elliptic equations with measure data

Cave

Durastanti

Oliva

2018

Nonlinear Differ. Equ. Appl.

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“…Also, we will always assume, without loss of generality, that µ d ≡ 0. Otherwise, in fact, the singularity of the problem virtually disappears and one is brought to nonsingular problem (see Remark 2.2 below, and the discussion in [17,Section 5]).…”

Section: Setting Of the Problem And Some Preliminary Tools 21 Main Amentioning

confidence: 99%

“…To deal with both existence and uniqueness for solutions to our problem we give the following variant of the definition of distributional solution to (2.1) given in [17] (and inspired by [36]).…”

Section: )mentioning

confidence: 99%

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

Oliva

Petitta

2018

Journal of Differential Equations

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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.

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“…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).…”

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confidence: 92%

Regularizing effect of absorption terms in singular problems

Oliva

2019

Journal of Mathematical Analysis and Applications

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A. We prove existence of solutions to problems whose model iswhere Ω is an open bounded subset of R N (N ≥ 2), ∆ is the -laplacian operator for 1 ≤ < N, > 0, γ ≥ 0 and is a nonnegative function in L (Ω) for some ≥ 1. In particular we analyze the regularizing effect produced by the absorption term in order to infer the existence of finite energy solutions in case γ ≤ 1. We also study uniqueness of these solutions as well as examples which show the optimality of the results. Finally, we find local W 1 -solutions in case γ > 1.

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Elliptic equations with general singular lower order term and measure data

Cited by 24 publications

References 13 publications

Existence and uniqueness results for possibly singular nonlinear elliptic equations with measure data

Existence and uniqueness results for possibly singular nonlinear elliptic equations with measure data

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

Regularizing effect of absorption terms in singular problems

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