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This paper deals with existence and multiplicity of positive solutions to the following class of nonlocal equations with critical nonlinearity:

In this article, we investigate multiplicity results for Choquard-Kirchhoff type equations, with Hardy-Littlewood-Sobolev critical exponents,

This paper deals with the existence of nontrivial solutions for critical Hardy–Schrödinger–Kirchhoff systems driven by the fractional p-Laplacian operator. Existence is derived as an application of the mountain pass theorem and the Ekeland variational principle. The main features and novelty of the paper are the presence of the Hardy terms as well as critical nonlinearities.

In this paper we derive and solve nonlocal elasticity a model describing the elastic behavior of composite materials, involving the fractional Laplacian operator. In dimension one we consider in (($\mathcal{D}$)) the case of a nonlocal elastic rod restrained at the ends, and we completely solve the problem showing the existence of a unique weak solution and providing natural sufficient conditions under which this solution is actually a classical solution of the problem. For the model (($\mathcal{D}$)) we also perform numerical simulations and a parametric analysis, in order to highlight the response of the rod, in terms of displacements and strains, according to different values of the mechanical characteristics of the material. The main novelty of this approach is the extension of the central difference method by the numerical estimate of the fractional Laplacian operator through a finite-difference quadrature technique. For higher dimensions N≥2{N\geq 2} we study more general problems for which the existence of weak solutions is proved via variational methods. The obtained results provide an original contribute in the knowledge of composite materials with properties of nonlocal elasticity.

The purpose of this paper is mainly to investigate the existence of entire solutions of the stationary Kirchhoff type equations driven by the fractional p-Laplacian operator in ℝN. By using variational methods and topological degree theory, we prove multiplicity results depending on a real parameter λ and under suitable general integrability properties of the ratio between some powers of the weights. Finally, existence of infinitely many pair of entire solutions is obtained by genus theory. Last but not least, the paper covers a main feature of Kirchhoff problems which is the fact that the Kirchhoff function M can be zero at zero. The results of this paper are new even for the standard stationary Kirchhoff equation involving the Laplace operator.

Many structural materials, which are preferred for the developing of advanced constructions, are inhomogeneous ones. Composite materials have complex internal structure and properties, which make them to be more effectual in the solution of special problems required for civil and environmental engineering. As a consequence of this internal heterogeneity, they exhibit complex mechanical properties. In this work, the analysis of some features of the behavior of composite materials under different loading conditions is carried out. The dependence of nonlinear elastic response of composite materials on loading conditions is studied. Several approaches to model elastic nonlinearity such as different stiffness for particular type of loadings and nonlinear shear stressstrain relations are considered. Instead of a set of constant anisotropy coeffcients, the anisotropy functions are introduced. Eventually, the combined constitutive relations are proposed to describe simultaneously two types of physical nonlinearities. The first characterizes the nonlinearity of shear stressstrain dependency and the latter determines the stress state susceptibility of material properties. Quite satisfactory correlation between the theoretical dependencies and the results of experimental studies is demonstrated, as described in [2, 3] as well as in the references therein. [ABSTRACT FROM AUTHOR]

The purpose of this paper is mainly to investigate the existence of weak solutions of fractional Laplacian equations with sublinear growth and oscillatory behavior. By using variational methods and abstract critical point theory, we want to establish the existence of multiplicity positive, multiplicity negative, and in particular, of multiplicity sign changing solutions. The multiplicity results depend on the real parameter. The key point is the choice of the framework to study the existence of weak solutions. In the suitable framework, by verifying that the conditions in abstract critical point theory are satisfied, we are able to fulfill our strategy. [ABSTRACT FROM AUTHOR]

In this paper we investigate the existence of multiple solutions for the nonhomogeneous fractional p-Laplacian equations of Schrödinger-Kirchhoff type in $${\mathbb {R}}^N$$ , where $$(-\varDelta )^s_p$$ is the fractional p-Laplacian operator, with $$0