We consider the following elliptic system ∆u = ∇H(u) in R N , where u : R N → R m and H ∈ C 2 (R m ), and prove, under various conditions on the nonlinearity H that, at least in low dimensions, a solution u = (ui) m i=1 is necessarily one-dimensional whenever each one of its components ui is monotone in one direction. Just like in the proofs of the classical De Giorgi's conjecture in dimension 2 (Ghoussoub-Gui) and in dimension 3 (Ambrosio-Cabré), the key step is a Liouville theorem for linear systems. We also give an extension of a geometric Poincaré inequality to systems and use it to establish De Giorgi type results for stable solutions as well as additional rigidity properties stating that the gradients of the various components of the solutions must be parallel. We introduce and exploit the concept of an orientable system, which seems to be key for dealing with systems of three or more equations. For such systems, the notion of a stable solution in a variational sense coincide with the pointwise (or spectral) concept of stability.
We classify finite Morse index solutions of the following nonlocal Lane-Emden equation (−∆) s u = |u| p−1 u R n for 1 < s < 2 via a novel monotonicity formula. For local cases s = 1 and s = 2 this classification is provided by Farina in [10] and Davila, Dupaigne, Wang and Wei in [8], respectively. Moreover, for the nonlocal case 0 < s < 1 finite Morse index solutions are classified by Davila, Dupaigne and Wei in [7].
Abstract. We are interested in the existence versus non-existence of nontrivial stable sub-and super-solutions ofwith positive smooth weights ω 1 (x), ω 2 (x). We consider the cases f (u) = e u , u p 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 ω 1 , ω 2 near infinity. Also the monotonicity of ω 1 is involved in some results. Our methods here are the methods developed by Farina, [9]. We examine a specific class of weights ω 1 (x) = (|x| 2 + 1) α 2 and ω 2 (x) = (|x| 2 + 1)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.
Abstract. We study stable solutions of the following nonlinear systemand Ω is a domain in R n . We introduce the novel notion of symmetric systems. The above system is said to be symmetric if the matrix of gradient of all components of H is symmetric. It seems that this concept is crucial to prove Liouville theorems, when Ω = R n , and regularity results, when Ω = B 1 , for stable solutions of the above system for a general nonlinearity H ∈ C 1 (R m ). Moreover, we provide an improvement for a linear Liouville theorem given in [20] that is a key tool to establish De Giorgi type results in lower dimensions for elliptic equations and systems.
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