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On the existence of solutions for Schrödinger-Maxwell systems in \(R^3\). (English) Zbl 1253.35166

Summary: In this paper we discuss the existence of solutions for the following Schrödinger-Maxwell systems \[ \begin{cases} -\Delta\psi+\lambda\psi+b(x)\phi\psi=a(x)|\psi|^{p-1}\psi \quad & \text{in} \, \mathbb R^3,\\ -\Delta\psi=4\pi b(x)\psi^2 \quad & \text{in} \, \mathbb R^3. \end{cases} \] Under suitable assumptions on \(a(x)\) and \(b(x)\), we establish existence results by variational methods.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q61 Maxwell equations
35J20 Variational methods for second-order elliptic equations
35J60 Nonlinear elliptic equations
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