Nonexistence theorems for some nonlinear systems of elliptic inequalities

We study the blow-up of solutions to two classes of nonlinear elliptic systems by using test function method.     

Liouville-type theorems for semilinear elliptic systems

In this paper, we provide two Liouville-type theorems for some semilinear elliptic systems. Our proofs are based on the moving spheres method and the general Kelvin transformed function. Four previously established results by different authors as consequence are obtained.     

Partial symmetry of vector solutions for elliptic systems

We consider partial symmetry of the least energy solutions of some nonlinear Schrödinger systems. We show that in the attractive case, all components of the least energy solutions possess foliated Schwarz symmetry with respect to the same point.     

Nonexistence of positive solutions to nonlinear nonlocal elliptic systems

In this paper we consider the question of nonexistence of nontrivial solutions for nonlinear elliptic systems involving fractional diffusion operators. Using a weak formulation approach and relying on a suitable choice of test functions, we derive sufficient conditions in terms of space dimension and systems parameters. Also, we present three main results associated to three different classes of systems.     

Nonexistence of nonnegative solutions of elliptic systems of divergence type

In this paper we deal with noncoercive elliptic systems of divergence type, that include both the p-Laplacian and the mean curvature operator and whose right-hand sides depend also on a gradient factor. We prove that any nonnegative entire (weak) solution is necessarily constant. The main argument of our proofs is based on previous estimates, given in Filippucci (2009) [12] for elliptic inequalities. Actually, the main technique for proving the central estimate has been developed by Mitidieri and Pohozaev (2001) [23] and relies on the method of test functions. No use of comparison and maximum principles or assumptions on symmetry or behavior at infinity of the solutions are required.     

Comparison theorems for nonlinear elliptic systems of second order with applications

In this paper comparison theorems concerning Dirichlet problem for a class of weakly coupled nonlinear second order elliptic systems have been developed. Some applications of these results to deflection or vibration of membranes and plates are also given.     

Nonexistence results and estimates for some nonlinear elliptic problems

Here we study the local or global behaviour of the solutions of elliptic inequalities involving quasilinear operators of the type or . We give integral estimates and nonexistence results. They depend on properties of the supersolutions of the equationsL _A ^u=0,L _B ^v=0, which suppose weak coercivity conditions. Under stronger conditions, we give pointwise estimates in case of equalities, using Harnack properties.     

Monotonicity and symmetry of solutions of elliptic systems in general domains

This work was done when the author visited Rutgers University under partial support of CNPq.     

Symmetry of components and Liouville theorems for noncooperative elliptic systems on the half-space

We study classical solutions of elliptic systems in the half-space and provide sufficient conditions for having symmetry (or proportionality) of components, i.e. u=Kv$u=Kv$ with K>0$K>0$, which then reduces the system to the scalar case. Under a natural structure condition on the nonlinearities, we show that solutions with sublinear growth, hence in particular bounded solutions, are symmetric. Noncooperative, nonvariational systems as well as supercritical nonlinearities can be covered. We also give an instance of our proportionality results without growth restriction on the solutions. As a consequence, we obtain new Liouville-type theorems in the half-space, as well as a priori estimates and existence results for related Dirichlet problems. Our proofs are based on a maximum principle, on the properties of suitable half-spherical means, on a rigidity result for superharmonic functions and on nonexistence of solution for scalar inequalities on the half-space.