## 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

### Citations:

Zbl 0541.49009; Zbl 0369.35022; Zbl 0453.47042
Full Text:

### References:

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