Existence results for non-local operators of elliptic type

In this paper, we investigate the existence of solutions for equations driven by a non-local integrodifferential operator with homogeneous Dirichlet boundary conditions. We make use of homological linking and Morse theory.     

Mountain pass type solutions for quasilinear elliptic equations

We establish the existence of weak solutions in an Orlicz-Sobolev space to the Dirichlet problem where is a bounded domain in , , and the function is an increasing homeomorphism from onto . Under appropriate conditions on , , and the Orlicz-Sobolev conjugate of (conditions which reduce to subcriticality and superlinearity conditions in the case the functions are given by powers), we obtain the existence of nontrivial solutions which are of mountain pass type. Received April 22, 1999 / Accepted June 11, 1999 / Published online April 6, 2000     

Liouville theorems for non-local operators

The paper characterizes some classes of pseudo-differential operators for which there are (or there are not) non-constant bounded harmonic functions. Non-local perturbations of Ornstein-Uhlenbeck operators and operators with dissipative coefficients are considered. The methods used are probabilistic and based on the concept of absorption function and on a new extension of the Bismut-Elworthy-Li formula. The probabilistic interpretation of the Liouville theorem by means of absorption functions for general Markov processes is given as well.     

Mountain Pass solutions for quasi-linear equations via a monotonicity trick

We obtain the existence of symmetric Mountain Pass solutions for quasi-linear equations without the typical assumptions which guarantee the boundedness of an arbitrary Palais-Smale sequence. This is done through a recent version of the monotonicity trick proved in Squassina (in press) [22]. The main results are new also for the p-Laplacian operator.     

Fundamental solutions of homogeneous elliptic differential operators

We compute fundamental solutions of homogeneous elliptic differential operators, with constant coefficients, on Rn by mean of analytic continuation of distributions. The result obtained is valid in any dimension, for any degree and can be extended to pseudodifferential operators of the same type.     

Coercivity for elliptic operators and positivity of solutions on Lipschitz domains

We show that usual second order operators in divergence form satisfy coercivity on Lipschitz domains if they are either complemented with homogeneous Dirichlet boundary conditions on a set of non-zero boundary measure or if a suitable Robin boundary condition is posed. Moreover, we prove the positivity of solutions in a general, abstract setting, provided that the right hand side is a positive functional. Finally, positive elements from W ^−1,2 are identified as positive measures.     

Two nontrivial solutions for hemivariational inequalities driven by nonlocal elliptic operators

In this paper, we establish the existence of two nontrivial solutions to a class of nonlocal hemivariational inequalities depending on two parameters. Our methods are based on critical point theory for non-differentiable functionals.     

Semilinear elliptic variational inequalities with dependence on the gradient via Mountain Pass techniques

In this paper we consider a semilinear variational inequality with a gradient-dependent nonlinear term. Obviously the nature of this problem is non-variational. Nevertheless we study that problem associating a suitable semilinear variational inequality, variational in nature, with it, and performing an iterative technique used in De Figueiredo et al. (2004) [6] in order to treat semilinear elliptic equations when there is a gradient dependence on the nonlinearity. We prove the existence of a non-trivial non-negative weak solution u for our problem using essentially variational methods, a penalization technique and an iterative scheme. Via Lewy-Stampacchia’s estimates and regularity theory for elliptic equation we also show that u is differentiable and its gradient is α-H?lder continuous on for any α∈(0,1).     

Harnack inequalities for non-local operators of variable order

We consider harmonic functions with respect to the operator

Under suitable conditions on we establish a Harnack inequality for functions that are nonnegative and harmonic in a domain. The operator is allowed to be anisotropic and of variable order.