Intertwining semiclassical bound states to a nonlinear magnetic Schrödinger equation

doi:10.1088/0951-7715/22/9/013

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Nonlinearity Volume 22 Number 9 Create an alert RSS this journal

Silvia Cingolani and Mónica Clapp 2009 Nonlinearity 22 2309 doi:10.1088/0951-7715/22/9/013

Intertwining semiclassical bound states to a nonlinear magnetic Schrödinger equation

Silvia Cingolani¹ and Mónica Clapp²

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Recommended by F Merle

We consider the magnetic NLS equation

$\begin{eqnarray} (-{\varepsilon \rmi}\nabla+A(x)) ^{2}u+V(x)u=\vert u\vert ^{p-2}u,\tqs x\in\mathbb{R}^{N}, \end{eqnarray}$

where N ≥ 3, 2 < p < 2^* := 2N/(N − 2), $A:\mathbb{R}^{N}\rightarrow \mathbb{R}^{N}$ is a magnetic potential and $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a bounded electric potential. We consider a group G of orthogonal transformations of $\mathbb{R}^{N}$ , and we assume that A(gx) = gA(x) and V(gx) = V(x) for any g G, $x\in\mathbb{R}^{N}$ . Given a group homomorphism $\tau:G\rightarrow\mathbb{S}^{1}$ into the unit complex numbers, we show the existence of semiclassical solutions $u_{\varepsilon}:\mathbb{R}^{N}\rightarrow\mathbb{C}$ to problem (0.1), which satisfy

$\begin{equation*} u_{\varepsilon}(gx)=\tau(g)u_{\varepsilon}(x) \end{equation*}$

for all g G, $x\in\mathbb{R}^{N}$ . Moreover, we show that there is a combined effect of the symmetries and the electric potential V on the number of solutions of this type.

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