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Multiple solutions to a magnetic nonlinear Choquard equation. (English) Zbl 1247.35141

The authors prove the existence of multiple complex valued solutions of the stationary magnetic nonlinear Choquard equation \[ \begin{split} (-i \nabla + A(x))^2 u + V(x)u = \left(\frac{1}{|x|^\alpha} \star |u|^p \right) |u|^{p-2} u, \\ u \in L^2( \mathbb{R}^N,\mathbb{C}), \enspace \nabla u + iA(x)u \in L^2(\mathbb{R}^N,\mathbb{C}^N),\end{split} \] where \(A: \mathbb{R}^N \rightarrow \mathbb{R}^N\) is a real-valued vector potential, \(V: \mathbb{R}^N \rightarrow \mathbb{R}\) is a real-valued scalar potential, \(N \geq 3\), \(\alpha \in (0,N)\), and \( 2 - (\alpha / N) < p < (2N - \alpha) / (N -2)\), for the case that both the vector and the scalar potential satisfy the symmetry conditions \(A(gx) = gA(x)\), \(V(gx) = V(x)\) for all \(g \in G, x \in \mathbb{R}^N\), where \(G\) is a closed subgroup of the group \(O(N)\) of linear isometries of \( \mathbb{R}^N\). Precisely, one finds solutions that satisfy the symmetry condition \(u(gx) = \tau(g) u(x)\), where \(\tau: G \rightarrow S^1\) is a known group holomorphism into the unit complex numbers \(S^1\). The main results are formulated in two theorems whose proofs are based on variational methods. For the lack of compactness problem they refer to P.-L. Lions [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1, 109–145 (1984; Zbl 0541.49009)].
This equation is a generalization of the stationary Choquard equation \[ -\Delta u + u = \left(\frac{1}{|x|} \star |u|^2 \right) u, \enspace u \in H^1(\mathbb{R}^3) \] that comes up in the Hartree-Fock theory of plasma. Existence and uniqueness results for this equation had been already established by E. H. Lieb [Studies Appl. Math. 57, 93–105 (1977; Zbl 0369.35022)] and P.-L. Lions [Nonlinear Anal., Theory Methods Appl. 4, 1063–1072 (1980; Zbl 0453.47042)].

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q40 PDEs in connection with quantum mechanics
35J20 Variational methods for second-order elliptic equations
35B06 Symmetries, invariants, etc. in context of PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35B40 Asymptotic behavior of solutions to PDEs
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