Qualitative properties of solutions to semilinear elliptic equations from the gravitational Maxwell Gauged O(3) Sigma model
Résumé
This article is devoted to the study of the following semilinear equation with measure data which originates in the gravitational Maxwell gauged O(3) sigma model, (E) −∆u + A 0 (k j=1 |x − p j | 2nj) −a e u (1 + e u) 1+a = 4π k j=1 n j δ pj − 4π l j=1 m j δ qj in R 2 , where {δ pj } k j=1 (resp. {δ qj } l j=1) are Dirac masses concentrated at the points {p j } k j=1 , (resp. {q j } l j=1), n j and m j are positive integers and a ≥ 0. We set N = k j=1 n j and M = l j=1 m j. In previous works [10, 31], some qualitative properties of solutions of (E) with a = 0 have been established. Our aim in this article is to study the more general case where a > 0. The additional difficulty of this case comes from the fact that the nonlinearity is no longer monotone and we cannot construct directly supersolutions and subsolutions anymore. Instead we develop a new and self-contained approach which enables us to emphasize the role played by the gravitation in gauged O(3) sigma model. Without the gravitational term, i.e. a = 0, problem (E) has a layer's structure of solutions {u β } β∈(−2(N −M), −2] , where u β is the unique non-topological solution such that u β = β ln |x| + O(1) for −2(N − M) < β < −2 and u −2 = −2 ln |x| − 2 ln ln |x| + O(1) at infinity respectively. On the contrary, when a > 0, the set of solutions to problem (E) has a much richer structure: besides the topological solutions, there exists a sequence of non-topological solutions in type I, i.e. such that u tends to −∞ at infinity, of non-topological solutions in type II, which tend to ∞ at infinity. The existence of these types of solutions depends on the values of the parameters N, M, β and on the gravitational interaction associated to a.
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