A remark on least energy solutions in RN

By Louis Jeanjean, Kazunaga Tanaka. 2002

330 citations recorded by the source corpus.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ea6408d387551dbdabdf52c49496d42b5fd4cb21

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Citations in the corpus

Existence of Groundstates for a Class of Nonlinear Choquard Equations in the Plane (2016, background): “rezis–Kato regularity had to be proved. The last ingredient that we need is an optimal path γv ∈ Γ associated to any solution v of (P). The construction of such paths (Proposition 5.3) is inspired by [10,17] but it is more delicate in our two-dimensional case than in the higher dimensions N ≥ 3, bec”
Singularly perturbed nonlinear elliptic problems on manifolds (2005, background): “Recently, Jeanjean and Tanaka [9] proved that a least energy solution U of (2) is a mountain pass solution.”
Standing Waves for Nonlinear Schrödinger Equations with a General Nonlinearity: One and Two Dimensional Cases (2008, methodology): “Also, for large R > 0 lim inf →0 ∫ B x R u 2dy ≥ 1 2 ∫ RN W̃ 2dy (36) Now, recalling from Jeanjean and Tanaka (2003a) that Ea > Eb if a > b, we see that EV x0 ≥ Em, since V x0 ≥ m.”
Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities (2009, background): “ Xk i=1 Em i . Thus, since Γε(u2,ε) ≥ o(1) we deduce from (3.22)-(3.23) that, for any i∈ {1,...k} (3.24) lim ε→0 Γε(ui 1,ε) = Em i . Now (3.22), (3.24) implies that LV(xi)(ωi) = Em i . Recalling from [19] that Ea >Eb if a>band using Lemma 2.11 we conclude that xi ∈ Mi. At this point it is clea”
Fractional Schrödinger–Poisson Systems with a General Subcritical or Critical Nonlinearity (2015, background): “As we can see in [26], b agrees with the least energy level of (3.”
Existence of groundstates for a class of nonlinear Choquard equations (2012, methodology): “The proof follows closely the arguments for the local problem developed by L. Jeanjean and K. Tanaka [20, lemma 2.1].”
A natural constraint approach to normalized solutions of nonlinear Schr\"odinger equations and systems (2016, methodology): “normalization constraint the Pohozaev manifold has been used in [39] in order to ﬁnd a ground state solution of a nonlinear Klein-Gordon equation. More recently the Pohozaev manifold has been used in [4,23,24] for investigating nonlinear (scalar) Schr¨odinger equations (without normalization constra”
A global branch approach to normalized solutions for the Schrödinger equation (2021, background): “Thus, recalling, see [23, 24] that, for any positive least action solution, there exists a path γ ∈ Γλ satisfying max t∈[0,1] Jλ(γ(t)) = mλ such that γ(t)(x) > 0 for all x ∈ R , t ∈ [0, 1], we deduce that the mountain pass characterization given in (6.”
Soliton solutions for quasilinear Schrodinger equations: the critical exponential case (2007, positive): “Now, in order to show c∞ ≤ I∞(ṽ), we slightly modify an argument used in [13] to get a path γ : [0, 1]→ H1(R2) such that   γ(0) = 0, I∞(γ(1)) < 0, ṽ ∈ γ([0, 1]), γ(t)(x) > 0 ∀x ∈ R2, t ∈ (0, 1], max t∈[0,1] I∞(γ(t)) = I∞(ṽ).”
Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential (2013, background): “It is known from [25] and [7, Proposition 2.”