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

Kirchhoff systems with dynamic boundary conditions

We are interested in the study of the global non-existence of solutions of hyperbolic nonlinear problems, governed by the p-Kirchhoff operator, under dynamic boundary conditions, when p > p_n with p_n < 2. The systems involve nonlinear external forces and may be affected by a perturbation. Several models already treated in the literature are covered in special subcases, and concrete examples are provided for the source term f and the external nonlinear boundary damping Q

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 L-K and involving a critical nonlinearity. In particular, we consider the problem -M(parallel to u parallel to(2))L(K)u = lambda f(x, u) + vertical bar u vertical bar(2s*-2) u in Omega, u = 0 in R-n \ Omega, where Omega subset of R-n is a bounded domain, 2(s)* is the critical exponent of the fractional Sobolev space H-s(R-n), the function f is a subcritical term and lambda is a positive parameter. The main feature, as well as the main difficulty, of the analysis is the fact that the Kirchhoff function M could 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. (C) 2015 Elsevier Ltd. All rights reserved.125699714Italian Project Caratterizzazione di modelli e sviluppo di codici di calcolo per il comportamento visco-termo-elastico di materiali compositi per l'edilizia sostenibile, l'efficienza energetica e la sostenibilita ambientale [UM12024L002, 10949]Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)INdAM-GNAMPA Project [Prot_2015_000368]MIUR [201274FYK7

Kirchhoff Systems with nonlinear source and boundary damping terms

Abstract. In this paper we treat the question of the non–existence of global solutions, or their long time behavior, of nonlinear hyperbolic Kirchhoff systems. The main p–Kirchhoff operator may be affected by a perturbation and the systems also involve an external force f and a nonlinear boundary damping Q. When p = 2, we consider some problems involving a higher order dissipation term, under dynamic boundary conditions. Special subcases of f and Q, interesting in applications, are presented in Sections 4, 5 and 6

Elliptic problems involving the fractional Laplacian in R^N

We study the existence and multiplicity of solutions for elliptic equations in R^N, driven by a non-local integro-differential operator, which main prototype is the fractional Laplacian. The model under consideration, denoted by (P), depends on a real parameter and involves two superlinear nonlinearities, one of which could be critical or even supercritical. The main theorem of the paper establishes the existence of three critical values of \lambda which divide the real line in different intervals, where (P) admits no solutions, at least one nontrivial non-negative entire solution and two nontrivial non-negative entire solutions

On the existence of stationary solutions for higher order p-Kirchhoff problems

In this paper we establish the existence of two nontrivial weak solutions of possibly degenerate nonlinear eigenvalue problems involving the p-polyharmonic Kirchhoff operator in bounded domains. The p-polyharmonic operators \Delta^L_p were recently introduced by Colasuonno and Pucci in 2011 for all orders L and independently in the same volume of the journal by Lubyshev only for L even. In Section 4 the results are then extended to non-degenerate p(x)-polyharmonic Kirchhoff operators. The main tool of the paper is a three critical points theorem given by Colasuonno, Pucci and Varga in 2012. Several useful properties of the underlying functional solution space [W^{L,p}_0(\Omega)]^d, endowed with the natural norm arising from the variational structure of the problem, are also proved both in the homogeneous case p=Const. and in the non-homogeneous case p=p(x). In the latter some sufficient conditions on the variable exponent p are given to prove the positivity of the the first eigenvalue of the p(x)-polyharmonic operator \Delta^L_{p(x)}

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 L-K and involving a critical nonlinearity. In particular, we consider the problem -M(parallel to u parallel to(2))L(K)u = lambda f(x, u) + vertical bar u vertical bar(2s*-2) u in Omega, u = 0 in R-n \ Omega, where Omega subset of R-n is a bounded domain, 2(s)* is the critical exponent of the fractional Sobolev space H-s(R-n), the function f is a subcritical term and lambda is a positive parameter. The main feature, as well as the main difficulty, of the analysis is the fact that the Kirchhoff function M could 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 literature125699714COORDENAÇÃO DE APERFEIÇOAMENTO DE PESSOAL DE NÍVEL SUPERIOR - CAPES33003017003P5-PNPD20131750-UNICAMP/MATEMATIC

Lifespan estimates for solutions of polyharmonic Kirchhoff systems

In mathematical physics we increasingly encounter PDEs models connected with vibration problems for elastic bodies and deformation processes, as it happens in the Kirchhoff-Love theory for thin plates subjected to forces and moments. Recently Monneanu proved in Refs. 26 and 27 the existence of a solution of the nonlinear Kirchhoff-Love plate model. In this paper we treat several questions about non-continuation for maximal solutions of polyharmonic Kirchhoff systems, governed by time-dependent nonlinear dissipative and driving forces. In particular, we are interested in the strongly damped Kirchhoff-Love model, containing also an intrinsic dissipative term of Kelvin-Voigt type. Global non-existence and a priori estimates for the lifespan of maximal solutions are proved. Several applications are also presented in special subcases of the source term f and the nonlinear external damping Q. \ua9 2012 World Scientific Publishing Company

SADDLE TYPE SOLUTIONS FOR A CLASS OF REVERSIBLE ELLIPTIC EQUATIONS

This paper is concerned with the existence of saddle type solutions for a class of semilinear elliptic equations of the type −∆u(x)+Fu(x,u) = 0, x ∈ Rn, n ≥ 2, (PDE) where F is a periodic and symmetric nonlinearity. Under a non degen- eracy condition on the set of minimal periodic solutions, saddle type solutions of (PDE) are found by a renormalized variational procedure

Asymptotic Stability for Anisotropic Kirchhoff Systems

We study the question of asymptotic stability, as time tends to infinity, of solutions of dissipative anisotropic Kirchhoff systems, involving the p(x)-Laplacian operator, governed by time-dependent nonlinear damping forces and strongly nonlinear power-like variable potential energies. This problem had been considered earlier for potential energies which arise from restoring forces, whereas here we allow also the effect of amplifying forces. Global asymptotic stability can then no longer be expected, and should be replaced by local stability. The results are further extended to the more delicate problem involving higher order damping terms