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Least action nodal solutions for the quadratic Choquard equation. (English) Zbl 1355.35079

Summary: We prove the existence of a minimal action nodal solution for the quadratic Choquard equation \[ -\Delta u + u = \bigl (I_\alpha \ast \| u\|^2\bigr )u \quad \text{in } \mathbb{R}^N, \] where \( I_{\alpha} \) is the Riesz potential of order \( \alpha \in (0,N)\). The solution is constructed as the limit of minimal action nodal solutions for the nonlinear Choquard equations \[ -\Delta u + u = \bigl (I_{\alpha} \ast \| u\|^p\bigr )| u|^{p-2}u \quad \text{in } \mathbb{R}^N \] when \( p\searrow 2\). The existence of minimal action nodal solutions for \( p>2\) can be proved using a variational minimax procedure over a Nehari nodal set. No minimal action nodal solutions exist when \( p<2\).

MSC:

35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
35J20 Variational methods for second-order elliptic equations
35Q55 NLS equations (nonlinear Schrödinger equations)
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