## 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 $$x_3\in\mathbb{R}^{N-2}$$.

### MSC:

 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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### References:

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