Solutions to nonlinear Schrödinger equations with singular electromagnetic potential and critical exponent. (English) Zbl 1273.35247

The authors consider singular (stationary) Schrödinger equations with a magnetic potential and a critical nonlinearity of the following form: \[ \left( \text{i}\nabla-\frac{A(\theta)}{|x|}\right)^2u -\frac{a}{|x|^2}u=|u|^{2^*-2}u \qquad\text{in }\mathbb{R}^N\setminus\{0\}. \tag{1} \] Here it is assumed that \(N\geq4\), and \(2^*:=2N/(N-2)\) denotes the usual critical Sobolev exponent. Moreover, \(A\in L^\infty(\mathbb{S}^{N-1},\mathbb{R}^N)\) is assumed to be equivariant under the action of \(G:= \text{SO}(2)\times \text{SO}(N-2)\). The main result states that there is \(a^*<0\) such that (1) possesses two solutions in \(D^{1,2}(\mathbb{R}^N)\) for every \(a<a^*\), one invariant under the action of \(G\) (i.e., biradially symmetric), and one invariant under \(\mathbb{Z}_k\times\text{SO}(N-2)\) for some \(k\in\mathbb{N}\).
An analogous result holds for magnetic Aharonov-Bohm type potentials of the form \[ \mathcal{A}(x_1,x_2,x_3) :=\left(\frac{-\alpha x_2}{x_1^2+x_2^2},\frac{\alpha x_1}{x_1^2+x_2^2},0\right), \] where \((x_1,x_2)\in\mathbb{R}^2\) and \(x_3\in\mathbb{R}^{N-2}\).


35Q55 NLS equations (nonlinear Schrödinger equations)
35J75 Singular elliptic equations
35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
35J20 Variational methods for second-order elliptic equations
35B06 Symmetries, invariants, etc. in context of PDEs
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