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

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.

### MSC:

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

Zbl 1434.35011
Full Text:

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