In this paper, we study the existence and the nonexistence of positive classical solutions of the static Hartree-Poisson equation −∆u = pu p−1 (|x| 2−n * u p), u > 0 in R n , where n ≥ 3 and p ≥ 1. The exponents of the Serrin type, the Sobolev type and the Joseph-Lundgren type play the critical roles as in the study of the Lane-Emden equation. First, we prove that the equation has no positive solution when 1 ≤ p < n+2 n−2 by means of the method of moving planes to the following system −∆u = √ pu p−1 v, u > 0 in R n , −∆v = √ pu p , v > 0 in R n. When p = n+2 n−2 , all the positive solutions can be classified as u(x) = c(t t 2 + |x − x * | 2) n−2 2

Hardy-Sobolev type integral equation Decay rates Integrable solution Bounded decaying solution In this paper, we are concerned with the Lane-Emden type 2m-order PDE with weightwhere n 3, p > 1, m ∈ [1, n/2), σ ∈ (−2m, 0], and the more general Hardy-Sobolev type integral equationthen there is not any positive solution to such an integral equation. Under the assumption of p > n+σ n−α , we obtain that the integrable solution u of the integral equation (i.e. u ∈ L n(p−1)α+σ (R n )) is bounded and decays fast with rate n − α. On the other hand, if the bounded solution is not integrable and decays with some rate, then the rate must be the slow one α+σ p−1 . In addition, the classical solution u of the 2m-order PDE satisfies the integral equation with α = 2m. Therefore, for the 2m-order PDE, all the decay properties above are still true.

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In this paper, we establish the sharp criteria for the nonexistence of positive solutions to the Hardy-Littlewood-Sobolev (HLS) type system of nonlinear equations and the corresponding nonlinear differential systems of Lane-Emden type equations. These nonexistence results, known as Liouville type theorems, are fundamental in PDE theory and applications. A special iteration scheme, a new shooting method and some Pohozaev type identities in integral form as well as in differential form are created. Combining these new techniques with some observations and some critical asymptotic analysis, we establish the sharp criteria of Liouville type for our systems of nonlinear equations. Similar results are also derived for the system of Wolff type integral equations and the system of γ-Laplace equations. A dichotomy description in terms of existence and nonexistence for solutions with finite energy is also obtained. Keywords: critical exponents, Liouville type theorems, HLS type integral equations, Wolff type integral equations, semilinear Lane-Emden equations, γ-Laplace equations, necessary and sufficient conditions of existence/nonexistence. MSC2000: 35J50, 45E10, 45G05 Other related work can be seen in [4], [31] and [32].Chen, Li and Ou [10] introduced the method of moving planes in integral forms to study the symmetry of the solutions for the HLS type system (1.1). Jin-Li and Hang thoroughly discussed the regularity of the solutions of (1.1) (cf.[18] and [20]). They found the optimal integrability intervals and established the smoothness for the integrable solutions. Based on the results, [27] gave the asymptotic behavior of the integrable solutions when |x| → 0 and |x| → ∞. Some Liouville type results can be seen in [2] and [6].Another significance of the work [10] is the equivalence of the integral equations and the PDEs involving the fractional order differential operator. Recently, the fractional Laplacians were applied extensively to describe various physical and finance phenomena, such as anomalous diffusion, turbulence and

Organoids are 3D structures grown from pluripotent stem cells derived from human tissue and serve as in vitro miniature models of human organs. Organoids are expected to revolutionize biomedical research and clinical care. However, organoids are not seen as morally neutral. For instance, tissue donors may perceive enduring personal connections with their organoids, setting higher bars for informed consent and patient participation. Also, several organoid sub-types, e.g., brain organoids and human–animal chimeric organoids, have raised controversy. This systematic review provides an overview of ethical discussions as conducted in the scientific literature on organoids. The review covers both research and clinical applications of organoid technology and discusses the topics informed consent, commercialization, personalized medicine, transplantation, brain organoids, chimeras, and gastruloids. It shows that further ethical research is needed especially on organoid transplantation, to help ensure the responsible development and clinical implementation of this technology in this field.