Uniform Bounds for Strongly Competing Systems: The Optimal Lipschitz Case

Abstract

For a class of systems of semi-linear elliptic equations, including (Formula presented.), for p = 2 (variational-type interaction) or p = 1 (symmetric-type interaction), we prove that uniform L^∞ boundedness of the solutions implies uniform boundedness of their Lipschitz norm as β→+∞, lthat is, in the limit of strong competition. This extends known quasi-optimal regularity results and covers the optimal case for this class of problems. The proofs rest on monotonicity formulae of Alt–Caffarelli–Friedman and Almgren type in the variational setting, and on the Caffarelli–Jerison–Kenig almost monotonicity formula in the symmetric one

A natural constraint approach to normalized solutions of nonlinear Schrödinger equations and systems

“…Kato argument we can check that uniform boundedness in H 1 (R 3 , R 2 ) implies also uniform boundedness in L ∞ (R 3 , R 2 ) (see[48, page 124] for a detailed proof, and[11] for the original argument). At this point, the rest of the proof follows directly by the general theory developed in[34,43,44,49].…”

Bose fluids and positive solutions to weakly coupled systems with critical growth in dimension two

“…We mention that for power-type nonlinearities, it turns out that actually 'reasonable' solutions for min{µ 1 , µ 2 } β max{µ 1 , µ 2 } do not exists, see [32] for the cubic case. Moreover, as β → −∞ one expects the appearance of segregation phenomena, in the sense that solutions concentrate on disjoint support, see for instance [12,13,18,35,38], or [34] for a recent survey on the subject. However, in dimension two and for exponential nonlinearities, analogous results seem to be out of reach at the moment, in particular due to the difficulty of obtaining existence results for β negative.…”

Competition in Periodic Media: III—Existence and Stability of Segregated Periodic Coexistence States

“…Thus, choosing β in such a way that S(x) ≤ 1 for x ∈ (−ℓε, ℓε), through a simple covering argument, the comparison principle yields where, as before, the constants B and C can be chosen independently of k and ε whenever k is sufficiently large. We can make use again a comparison with a super-solution, see [20,Lemma 2.2], and conclude that With the uniform estimates of Lemma 3.8 and Lemma 3.9 we are now in position to show that the solution (u 1 , u 2 ) constructed in the previous section is indeed linearly stable if k is sufficiently large.…”