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In this paper we first present the classical maximum principle due to E. Hopf, together with an extended commentary and discussion of Hopf's paper. We emphasize the comparison technique invented by Hopf to prove this principle, which has since become a main mathematical tool for the study of second order elliptic partial differential equations and has generated an enormous number of important applications. While Hopf's principle is generally understood to apply to linear equations, it is in fact also crucial in nonlinear theories, such as those under consideration here.In particular, we shall treat and discuss recent generalizations of the strong maximum principle, and also the compact support principle, for the case of singular quasilinear elliptic differential inequalities, under generally weak assumptions on the quasilinear operators and the nonlinearities involved. Our principal interest is in necessary and sufficient conditions for the validity of both principles; in exposing and simplifying earlier proofs of corresponding results; and in extending the conclusions to wider classes of singular operators than previously considered.The results have unexpected ramifications for other problems, as will develop from the exposition, e.g.(i) two point boundary value problems for singular quasilinear ordinary differential equations (Sections 3 and 4); (ii) the exterior Dirichlet boundary value problem (Section 5); (iii) the existence of dead cores and compact support solutions, i.e. dead cores at infinity (Section 7); (iv) Euler-Lagrange inequalities on a Riemannian manifold (Section 9); (v) comparison and uniqueness theorems for solutions of singular quasilinear differential inequalities (Section 10).

In this paper we investigate the existence of multiple solutions for the nonhomogeneous fractional p-Laplacian equations of Schrödinger-Kirchhoff typeand ps < N , the nonlinearity f : R N × R → R is a Carathéodory function and satisfies the Ambrosetti-Rabinowitz condition, V : R N → R + is a potential function and g : R N → R is a perturbation term. We first establish Batsch-Wang type compact embedding theorem for the fractional Sobolev spaces. Then multiplicity results are obtained by using the Ekeland variational principle and the Mountain Pass theorem. Mathematics Subject Classification

The purpose of this paper is mainly to investigate the existence of entire solutions of the stationary Kirchhoff type equations driven by the fractional

Vazquez in 1984 established a strong maximum principle for the classical m-Laplace differential inequality ∆ m u − f (u) ≤ 0, where ∆ m u =div(|Du| m−2 Du) and f (u) is a non-decreasing continuous function with f (0) = 0. We extend this principle to a wide class of singular inequalities involving quasilinear divergence structure elliptic operators, and also consider the converse problem of compact support solutions in exterior domains.

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Using the Mountain-Pass Theorem of Ambrosetti and Rabinowitz we prove thatThe technique is based on the existence of extremals of some Hardy-Sobolev type embeddings of independent interest. We also show that if u ∈ D p 1 (R n ) is a weak solution in R n of −∆pu − µ|x| −p |u| p−2 u = |x| −s |u| p ⋆ (s)−2 u + |u| q−2 u, then u ≡ 0 when either 1 < q < p ⋆ , or q > p ⋆ and u is also of class L ∞ loc (R n \ {0}).