Quasilinear Lane-Emden equations with absorption and measure data

Bidaut-Véron, Marie-Françoise; Quốc, Hùng Nguyễn; Верон, Лаурент

doi:10.48550/arxiv.1212.6314

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“…M(Ω, ρ 0 ) = M b (Ω) is the set of bounded Radon measures and g is nondecreasing and satisfies the β-subcritical assumption: The study of general semilinear elliptic equations with measure data have been investigated, such as the equations involving measures boundary data which was initiated by Gmira and Véron [20] who adapted the method introduced by Benilan and Brezis to obtain the existence and uniqueness of solution. This subject has been vastly expanded in recent years, see the papers of Marcus and Véron [25,26,27,28], Bidaut-Véron and Vivier [5], Bidaut-Véron, Hung and Véron [4]. Recently, great attention has been devoted to non-linear equations involving fractional Laplacian or more general integro-differential operators and we mention the reference [8,9,10,14,15,24,30,32].…”

Section: Introductionmentioning

confidence: 99%

Semilinear fractional elliptic equations involving measures

Chen

Верон

2014

Journal of Differential Equations

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We study the existence of weak solutions to (E) (−∆) α u + g(u) = ν in a bounded regular domain Ω in R N (N ≥ 2) which vanish in R N \ Ω, where (−∆) α denotes the fractional Laplacian with α ∈ (0, 1), ν is a Radon measure and g is a nondecreasing function satisfying some extra hypotheses. When g satisfies a subcritical integrability condition, we prove the existence and uniqueness of a weak solution for problem (E) for any measure. In the case where ν is Dirac measure, we characterize the asymptotic behavior of the solution. When g(r) = |r| k−1 r with k supercritical, we show that a condition of absolute continuity of the measure with respect to some Bessel capacity is a necessary and sufficient condition in order (E) to be solved.

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