Semilinear fractional elliptic equations involving measures

We study the existence of weak solutions to (E) (−Δ)^αu+g(u)=ν${(-\mathrm{\Delta })}^{\alpha }u+g(u)=\nu$ in a bounded regular domain Ω   in R^N(N≥2)${\mathbb{R}}^N(N\ge 2)$ which vanish in R^N?Ω${\mathbb{R}}^N?\Omega$, where (−Δ)^α${(-\mathrm{\Delta })}^{\alpha }$ denotes the fractional Laplacian with α∈(0,1)$\alpha \in (0,1)$, 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 weak solution for problem (E) for any measure. In the case where ν   is a Dirac measure, we characterize the asymptotic behavior of the solution. When g(r)=|r|^k−1r$g(r)={|r|}^{k-1}r$ 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.     

Quasilinear Lane–Emden equations with absorption and measure data

We study the existence of solutions to the equation −_Δpu+g(x,u)=μ$-{\mathrm{\Delta }}_{p}u+g(x,u)=\mu$ when g(x,.)$g(x,.)$ is a nondecreasing function and μ   a measure. We characterize the good measures, i.e. the ones for which the problem has a renormalized solution. We study particularly the cases where g(x,u)=|x|^−β|u|^q−1u$g(x,u)={|x|}^{-\beta }{|u|}^{q-1}u$ and g(x,u)=sgn(u)(e^τ|u|λ−1)$g(x,u)=\mathrm{sgn}(u)(e^{\tau {|u|}^{\lambda }}-1)$. The results state that a measure is good if it is absolutely continuous with respect to an appropriate Lorentz–Bessel capacities.