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.     

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.     

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.     

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).     

Eigenvalue bounds for elliptic operators

Lower bounds for the real parts of the points in the spectrum of elliptic equations are derived. These bounds, depending only on the diameter L of the domain G and on the maximum norm M of the coefficients a, b, are optimal. They are always positive and thus the spectrum is bounded away from the imaginary axis. This result is then used to prove an “anti-dynamo theorem” for magnetic fields with plane symmetry in the case of a compressible steady flow surrounded by a perfect conductor.     

A Mountain Pass for Reacting Molecules

In this paper, we consider a neutral molecule that possesses two distinct stable positions for its nuclei, and look for a mountain pass point between the two minima in the non-relativistic Schrödinger framework. We first prove some properties concerning the spectrum and the eigenstates of a molecule that splits into pieces, a behavior which is observed when the Palais-Smale sequences obtained by the mountain pass method are not compact. This enables us to identify precisely the possible values of the mountain pass energy and the associated critical points at infinity (a concept introduced by Bahri [2]) in this non-compact case. We then restrict our study to a simplified (but still relevant) model: a molecule made of two interacting parts, the geometry of each part being frozen. We show that this lack of compactness is impossible under some natural assumptions about the configurations at infinity, proving the existence of the mountain pass in these cases. More precisely, we suppose either that the molecules at infinity are charged, or that they are neutral but with dipoles at their ground state. Communicated by Rafael D. Benguria Submitted 07/07/03, accepted 27/01/04     

Spectral properties of non-local differential operators

In this article we consider the spectral properties of a class of non-local operators that arise from the study of non-local reaction-diffusion equations. Such equations are used to model a variety of physical and biological systems with examples ranging from Ohmic heating to population dynamics. The operators studied here are bounded perturbations of linear (local) differential operators. The non-local perturbation is in the form of an integral term. It is shown here that the spectral properties of these non-local operators can differ considerably from those of their local counterpart. Multiplicities of eigenvalues are studied and new oscillation results for the associated eigenfunctions are presented. These results highlight problems with certain similar results and provide an alternative formulation. Finally, the stability of steady states of associated non-local reaction-diffusion equations is discussed.     

Geometry of phase plane and radial solutions for nonlinear elliptic equations with extremal operators

We study the existence of positive radially symmetric solutions to a class of nonlinear elliptic problems involving extremal operators and nonlinearity of exponential or polynomial type. According to the values of a parameter, we describe situations where the equation has one, finitely many, infinitely many or no solutions, by using the geometry structure of phase plane.