A natural constraint approach to normalized solutions of nonlinear Schrödinger equations and systems. (English) Zbl 1485.35173

Bartsch, Thomas; Soave, Nicola

J. Funct. Anal. 272, No. 12, 4998-5037 (2017); correction ibid. 275, No. 2, 516-521 (2018).

The paper [J. Funct. Anal. 272, No. 12, 4998–5037 (2017; Zbl 06714264)] of the authors concerns the existence of positive normalized solutions to the semilinear elliptic system \[\begin{split} -\Delta u_1 - \lambda _1u_1 &= \mu_1u^3_1 + \beta u_2u_1, \\ -\Delta u_2 - \lambda_2u_2 &= \mu_2 u^3_2 + \beta u^2_ 1u_2, \qquad \text{in}\ \mathbb{R}^3, \end{split} \tag{a}\] i.e., the existence of real numbers \((\lambda_1, \lambda_2)\in\mathbb{R}^2\) and of functions \((u_1, u_2)\in H^1(\mathbb{R}^3; \mathbb{R}^2)\) satisfying (a) joint with the normalization condition \[ \int_{\mathbb{R}^3} u^2_1 = a^2_1,\qquad \text{and}\quad \int_{\mathbb{R}^3} u^2_2 = a^2 _2 ,\tag{b} \] for a priori given \(a_1, a_2 > 0,\ \mu _1,\mu_2, \beta\in\mathbb{R}.\)
The authors look for solutions to (a)–(b) as critical points of the corresponding energy functional on the constraint set defined keeping (b) fixed, and beeing \((\lambda_1,\lambda_2)\) the Lagrange multipliers. In the literature there are already known several particular cases of existence of solutions (\((a_1,a_2)\) small enough, restrictions on the variations of the parameters, \(\ldots\))
In their Theorem 1.1, the authors prove the existence of positive, radially symmetric solutions for \(\lambda_i<0\), when \(\mu_ 1,\mu_ 2,a_1,a_2 > 0\) and \(\beta< 0\) is fixed.
In their Theorem 1.2, keeping \(a_1,a_2, \mu_1\) and \(\mu_2\) fixed, the authors prove the existence of a family of solutions \[ \{(\lambda_{1,\beta} ,\lambda_{2,\beta} ,\overline{u}_{1,\beta} ,\overline{u}_{2,\beta} ) : \beta < 0\} \] and prove that phase-separation occurs as \(\beta \to -\infty.\) Specifically (up to a subsequence):
(i) \((\lambda_{1,\beta} ,\lambda_{2,\beta} ) \to (\lambda_1,\lambda_2)\), with \(\lambda_1,\lambda_2 \le 0\);
(ii) \((\overline{u}_{1,\beta} ,\overline{u}_{2,\beta} ) \to (\overline{u}_1 ,\overline{u}_2 )\) in \(C^{0,\alpha}_{\mathrm{loc}}(\mathbb{R}^3)\) and in \(H^1_{\mathrm{loc}}(\mathbb{R}^3)\);
(iii) \(\overline{u}_1 \) and \(\overline{u}_2 \) are nonnegative Lipschitz continuous functions having disjoint positivity sets, in the sense that \(\overline{u}_1 \overline{u}_2 \equiv 0\) in \(\mathbb{R}^3\);
(iv) the difference \(\overline{u}_1 - \overline{u}_2 \) is a sign-changing radial solution of \[ - \Delta w -\lambda_ 1 w^+ +\lambda_ 2 w^- = \mu_1 (w_1^+)^3 - \mu_2 (w_1^-)^3 \qquad\text{in}\quad \mathbb{R}^3 . \]
After the publication, Prof. J. Mederski pointed out that the proofs of [loc. cit., Theorem 2.1(i) and Theorem 4.1(i)] contain a gap. In this correction note the authors show that the main results from [loc. cit.] are correct as stated, and give modified statements, slightly weaker than previous subsidiary Theorems, to get this.

Reviewer: Rosa Maria Pardo San Gil (Madrid)

MSC:

35J47	Second-order elliptic systems
35J50	Variational methods for elliptic systems
35J91	Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian

Keywords:

elliptic systems; Schrödinger systems; natural constraint; min-max methods

Citations:

Zbl 1434.35011

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