Minimal Mass Blow-Up Solutions for the \(L²\)-Critical NLS with the Delta Potential for Even Data in One Dimension

.We consider the \(L^2\)-critical nonlinear Schrödinger equation (NLS) with the delta potential \(i∂_tu +∂^2_x u + μ δ u +|u|⁴u=0, \, \, t∈ R, \, x∈ R,\) where \( μ ∈ R\) and \(δ\) is the Dirac delta distribution at \(x=0\). Local well-posedness theory, together with the sharp Gagliardo–Nirenberg inequality and the conservation laws of mass and energy, implies that the solution with mass less than \(\|Q\|₂\) is global existence in \(H^1(R)\), where \(Q\) is the ground state of the \(L^2\)-critical NLS without the delta potential (i.e., \(μ=0\)). We are interested in the dynamics of the solution with threshold mass \(\|u_0\|₂=\|Q\|₂\) in \(H^1(R)\). First, for the case \(μ=0\), such a blow-up solution exists due to the pseudoconformal symmetry of the equation and is unique up to the symmetries of the equation in \(H^1(R)\) from Merle [Duke Math. J., 69 (1993), pp. 427–454] and recently in \(L^2(R)\) from Dodson [arXiv:2104.11690, 2021]. Second, for the case \(μ < 0\), a simple variational argument with the conservation laws of mass and energy implies that even solutions with threshold mass exist globally in \(H^1(R)\). Finally, for the case \(μ > 0\), we show the existence of even threshold solutions with blow-up speed determined by the sign (i.e., \(μ > 0\)) of the delta potential perturbation since the refined blow-up profile to the rescaled equation is stable in a precise sense. The key ingredients here, including the Energy–Morawetz argument and the compactness method as well as modulation analysis, are close to the original one in Raphaël and Szeftel [J. Amer. Math. Soc., 24 (2011), pp. 471–546].Keywordsblow-upconcentration-compactness argumentcompactness methodDirac delta potentialEnergy–Morawetz estimatemodulation analysisnonlinear Schrödinger equationMSC codes35Q5535B44