Mostafa Fazly

Liouville Type Theorems for Stable Solutions of Certain Elliptic Systems

Abstract We establish Liouville type theorems for elliptic systems with various classes of nonlinearities on ℝN. We show, among other things, that a system has no semi-stable solution in any dimension, whenever the infimum of the derivatives of the corresponding non-linearities is positive. We give some immediate applications to various standard systems, such as the Gelfand, and certain Hamiltonian systems. The case where the infimum is zero is more interesting and quite challenging. We show that any C2(ℝN) positive entire semi-stable solution of the following Lane-Emden system, is necessarily constant, whenever the dimension N < 8 + 3α + , provided p = 1, q ≥ 2 and f (x) = (1 + |x|2). The same also holds for p = q ≥ 2 provided We also consider the case of bounded domains Ω ⊂ ℝN, where we extend results of Brown et al. [1] and Tertikas [18] about stable solutions of equations to systems. At the end, we prove a Pohozaev type theorem for certain weighted elliptic systems.

On stable entire solutions of semi-linear elliptic equations with weights

We are interested in the existence versus non-existence of nontrivial stable suband super-solutions of (0.1) −div(ω1∇u) = ω2f(u) in R , with positive smooth weights ω1(x), ω2(x). We consider the cases f(u) = eu, up where p > 1 and −u−p where p > 0. We obtain various non-existence results which depend on the dimension N and also on p and the behaviour of ω1, ω2 near infinity. Also the monotonicity of ω1 is involved in some results. Our methods here are the methods developed by Farina. We examine a specific class of weights ω1(x) = (|x|2 + 1) α 2 and ω2(x) = (|x|2 + 1) β 2 g(x), where g(x) is a positive function with a finite limit at ∞. For this class of weights, non-existence results are optimal. To show the optimality we use various generalized Hardy inequalities.

Symmetry Results for Fractional Elliptic Systems and Related Problems

We study elliptic gradient systems with fractional laplacian operators on the whole space where u: R n → R m , H ∈ C 2, γ(R m ) for γ > max (0, 1 − 2 min {s i }), s = (s 1,…, s m ) for 0 < s i < 1 and ∇H(u) = (H u i (u 1, u 2,…, u m )) i . We prove De Giorgi type results for this system for certain values of s and in lower dimensions, i.e. n = 2, 3. Just like the local case, the concepts of orientable systems and H − monotone solutions play the key role in proving symmetry results. In addition, we provide optimal energy estimates, a monotonicity formula, a Hamiltonian identity and various Liouville theorems.