×

zbMATH — the first resource for mathematics

Asymptotics for \(L^ 2\) minimal blow-up solutions of critical nonlinear Schrödinger equation. (English) Zbl 0862.35013
From the author’s introduction: We describe the behaviour of a sequence \(v_n:\mathbb{R}^N\to\mathbb{C}\) of \(H^1\) functions such that \[ \int|v_n|^2=\int Q^2,\;E(v_n)={1\over 2}\int|\nabla v_n|^2-{1\over{4\over N}+2}\int|v_n|^{{4\over N}+2}\leq E_0,\;\int|\nabla v_n|^2\to+\infty, \] where \(Q\) is the positive radial symmetric solution of the equation \(\Delta v+|v|^{4/N}v=v\).

MSC:
35B40 Asymptotic behavior of solutions to PDEs
35J60 Nonlinear elliptic equations
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] Berestycki, H.; Lions, P. L., Nonlinear scalar field equations I. existence of a ground state; II. existence of infinitely many solutions, Arch. Rational Mech. Anal., Vol. 82, 313-375, (1983)
[2] Ginibre, J.; Velo, G., On a class of nonlinear Schrödinger equations I, II. the Cauchy problem, general case, J. Funct. Anal., Vol. 32, 1-71, (1979) · Zbl 0396.35028
[3] Kato, T., On nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré, Physique Théorique, Vol. 49, 113-129, (1987) · Zbl 0632.35038
[4] Kwong, M. K., Uniqueness of positive solutions of δuu + u^P = 0 in ℝ^N, Arch. Rational Mech. Anal., Vol. 105, 243-266, (1989) · Zbl 0676.35032
[5] Merle, F., Determination of blow-up solutions with minimal mass for Schrödinger equation with critical power, Duke J., Vol. 69, 427-454, (1993) · Zbl 0808.35141
[6] F. Merle, Nonexistence of minimal blow-up solutions of equations iu_t = − Δu - k(x)|u|^4/Nu in ℝ^N, preprint. · Zbl 0846.35129
[7] Strauss, W. A., Existence of solitary waves in higher dimensions, Commun. Math. Phys., Vol. 55, 149-162, (1977) · Zbl 0356.35028
[8] Weinstein, M. I., Modulational stability of ground states of the nonlinear Schrödinger equations, SIAM J. Math. Anal., Vol. 16, 472-491, (1985) · Zbl 0583.35028
[9] Weinstein, M. I., Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys., Vol. 87, 567-576, (1983) · Zbl 0527.35023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.