A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system

Abstract

The paper is concerned with the local and global bifurcation structure of positiveThe system arises in nonlinear optics and in the Hartree-Fock theory for a double condensate. Local and global bifurcations in terms of the nonlinear coupling parameter β of the system are investigated by using spectral analysis and by establishing a new Liouville type theorem for nonlinear elliptic systems which provides a-priori bounds of solution branches. If the domain is radial, possibly unbounded, then we also control the nodal structure of a certain weighted difference of the components of the solutions along the bifurcating branches. Mathematics Subject Classification (2000)35B05 · 35B32 · 35J50 · 35J55 · 58C40 · 58E07 Dedicated to Paul Rabinowitz on the occasion of his 70th birthday.

Positive Least Energy Solutions and Phase Separation for Coupled Schrödinger Equations with Critical Exponent

“…2) In the case where N 3, the nonlinearity and the coupling terms in (1.2) are subcritical [that is, 4 < 2N /(N − 2)], and the existence of solutions has recently received great interest, see [3,4,10,11,20,24,29] for the existence of a (least energy) solution, [21,22,25,28] for semiclassical states or singularly perturbed settings, and [7,17,23,26,32,33] for the existence of multiple solutions.…”

Semi-classical standing waves for nonlinear Schrödinger systems

“…Furthermore, Morse indices of scalar solutions (U 1 , 0), (0, U 2 ) of (5) go to ∞ as β → ∞. This shows quite contrasted characteristics between vector solutions and scalar solutions for the limiting problem (5).…”

“…On the other hand, through the initial works [24,25,33], it is known in [1] that there exist 0 < β 1 , β 2 < ∞ [see (11), (12) for the definition] depending on W 1 (y), W 2 (y), μ 1 , μ 2 , such that for β ∈ (0, min{β 1 , β 2 }), there exists a vector solution U y given by the mountain pass theorem on the Nehari manifold {u ∈ (H 1 r ) 2 \{(0, 0)} | ∇ u L(y, u)u = 0} and for β ∈ (max{β 1 , β 2 }, ∞), there exists a vector solution U y given by the mountain pass theorem on the whole space (H 1 r ) 2 , which is a least energy solutions among all nontrivial solutions of (5). This implies that Morse index of the vector solution is 2 for β ∈ (0, min{β 1 , β 2 }) and 1 for β ∈ (max{β 1 , β 2 }, ∞).…”

A local mountain pass type result for a system of nonlinear Schrödinger equations

“…There are many works on the existence of non-trivial positive solutions of (1.5)- (1.8). See [2][3][4][5]11,12,[18][19][20]24,26,30,33,34]. Sign and size of β are important in the study of (1.5)-(1.8) and various situations are studied in the above papers.…”