Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity

This paper deals with the existence and the asymptotic behavior of non-negative solutions for a class of stationary Kirchhoff problems driven by a fractional integro-differential operator $\mathcal L_K$ and involving a critical nonlinearity. The main feature, as well as the main difficulty, of the analysis is the fact that the Kirchhoff function $M$ can be zero at zero, that is the problem is degenerate. The adopted techniques are variational and the main theorems extend in several directions previous results recently appeared in the literature

Entire solutions for critical p-fractional Hardy Schrodinger Kirchhoff equations

Existence theorems of nonnegative entire solutions of stationary critical p-fractional Hardy Schr¨odinger Kirchhoff equations are presented in this paper. The equations we treat deal with Hardy terms and critical nonlinearities and the main theorems extend several recent results on the topic. The paper contains also some open problems

The strong maximum principle revisited

AbstractIn 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). The case of p-regular elliptic inequalities is briefly considered in Section 11

A note on the strong maximum principle for elliptic differential inequalities

AbstractWe consider the strong maximum principle and the compact support principle for quasilinear elliptic differential inequalities, under generally weak assumptions on the quasilinear operators and the nonlinearities involved. This allows us to give necessary and sufficient conditions for the validity of both principles

(p,Q) systems with critical singular exponential nonlinearities in the Heisenberg group

AbstractThe paper deals with the existence of solutions for(p,Q)(p,Q)coupled elliptic systems in the Heisenberg group, with critical exponential growth at infinity and singular behavior at the origin. We derive existence of nonnegative solutions with both components nontrivial and different, that is solving an actual system, which does not reduce into an equation. The main features and novelties of the paper are the presence of a general coupled critical exponential term of the Trudinger-Moser type and the fact that the system is set inℍn{{\mathbb{H}}}^{n}

Existence for (p, q) critical systems in the Heisenberg group

Abstract This paper deals with the existence of entire nontrivial solutions for critical quasilinear systems () in the Heisenberg group ℍn, driven by general (p, q) elliptic operators of Marcellini types. The study of () requires relevant topics of nonlinear functional analysis because of the lack of compactness. The key step in the existence proof is the concentration–compactness principle of Lions, here proved for the first time in the vectorial Heisenberg context. Finally, the constructed solution has both components nontrivial and the results extend to the Heisenberg group previous theorems on quasilinear (p, q) systems