## 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:

  L. Abatangelo and S. Terracini, A note on the complete rotational invariance of biradial solutions to semilinear elliptic equations. Preprint, 2009. · Zbl 1218.35105  Arioli G., Szulkin A.: A semilinear Schrödinger equation in the presence of a magnetic field. Arch. Ration. Mech. Anal. 170, 277–295 (2003) · Zbl 1051.35082  Bartsch T., Dancer E.N., Peng S.: On multi-bump semi-classical bound states of nonlinear Schrödinger equations with electromagnetic fields. Adv. Differential Equations 11, 781–812 (2006) · Zbl 1146.35081  Bahri A., Li Y.Y.: On a min-max procedure for the existence of positive solutions for certain scalar field equations in $${$$\backslash$$mathbb{R}\^N}$$ . Rev. Mat. Iberoamericana 6, 1–15 (1990) · Zbl 0731.35036  Bourgain J., Brezis H.: New estimates for elliptic equations and Hodge type systems. J. Eur. Math. Soc. (JEMS) 9(2), 277–315 (2007) · Zbl 1176.35061  Brezis H., Nirenberg L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm. Pure Appl. Math. 36, 437–477 (1983) · Zbl 0541.35029  Chabrowski J., Szulkin A.: On the Schrödinger equation involving a critical Sobolev exponent and magnetic field. Topol. Methods Nonlinear Anal. 25, 3–21 (2005) · Zbl 1176.35022  Cingolani S.: Semiclassical stationary states of nonlinear Schrödinger equations with an external magnetic field. J. Differential Equations 188, 52–79 (2003) · Zbl 1062.81056  Cingolani S., Clapp M.: Intertwining semiclassical bound states to a nonlinear magnetic Schrödinger equation. Nonlinearity 22, 2309–2331 (2009) · Zbl 1173.35678  Cingolani S., Secchi S.: Semiclassical limit for nonlinear Schrödinger equations with electromagnetic fields. J. Math. Anal. Appl. 275, 108–130 (2002) · Zbl 1014.35087  Cingolani S., Secchi S.: Semiclassical states for NLS equations with magnetic potentials having polynomial growths. J. Math. Phys. 46, 053503 (2005) · Zbl 1110.81081  Clapp M., Iturriaga R., Szulkin A.: Periodic and Bloch solutions to a magnetic nonlinear Schrödinger equation. Adv. Nonlinear Stud. 9, 639–655 (2009) · Zbl 1182.35203  Clapp M., Szulkin A.: Multiple solutions to a nonlinear Schrödinger equation with Aharonov-Bohm magnetic potential. Nonlinear Differential Equations Appl. 17, 229–248 (2010) · Zbl 1189.35302  L. C. Evans, Partial differential equations. 2nd ed., Grad. Stud. Math. 19, American Mathematical Society, Providence, RI, 2010. · Zbl 1194.35001  M. Esteban and P. L. Lions, Stationary solutions of nonlinear Schrödinger equations with an external magnetic field. In: Partial Differential Equations and the Calculus of Variations, Vol. I, Progr. Nonlinear Differential Equations Appl. 1, Birkhäuser Boston, Boston, MA, 1989, 401–449.  Felli V., Ferrero A., Terracini S.: Asymptotic behavior of solutions to Schrödinger equations near an isolated singularity of the electromagnetic potential. J. Eur. Math. Soc. (JEMS) 13, 119–174 (2011) · Zbl 1208.35070  Felli V., Marchini E.M., Terracini S.: On the behavior of solutions to Schrödinger equations with dipole-type potentials near the singularity. Discrete Contin. Dyn. Syst. 21, 91–119 (2008) · Zbl 1141.35362  Kurata K.: A unique continuation theorem for the Schrödinger equation with singular magnetic field. Proc. Amer. Math. Soc. 125, 853–860 (1997) · Zbl 0887.35026  Kurata K.: Existence and semi-classical limit of the least energy solution to a nonlinear Schrödinger equation with electromagnetic fields. Nonlinear Anal. 41, 763–778 (2000) · Zbl 0993.35081  A. Laptev and T. Weidl, Hardy inequalities for magnetic Dirichlet forms. In: Mathematical Results in Quantum Mechanics (Prague, 1998), Oper. Theory Adv. Appl. 108, Birkhäuser, Basel, 1999, 299–305. · Zbl 0977.26005  Leinfelder H.: Gauge invariance of Schrödinger operators and related spectral properties. J. Operator Theory 9, 163–179 (1983) · Zbl 0528.35024  Pankov A.A.: On nontrivial solutions of the nonlinear Schrödinger equation with a magnetic field. Funct. Anal. Appl. 37, 75–77 (2003) · Zbl 1028.35142  G. Rozenblum and M. Melgaard, Schrödinger operators with singular potentials. In: Stationary Partial Differential Equations, Vol. II, Handb. Differ. Equ., Elsevier/North–Holland, Amsterdam, 2005, 407–517. · Zbl 1206.35084  Reed M., Simon B.: Methods of modern mathematical physics. II. Fourier analysis, self-adjointness. Academic Press, New York (1975) · Zbl 0308.47002  Schindler I., Tintarev K.: A nonlinear Schrödinger equation with external magnetic field. Rostock. Math. Kolloq. 56, 49–54 (2002) · Zbl 1140.35567  Solimini S.: A note on compactness-type properties with respect to Lorentz norms of bounded subsets of a Sobolev space. Ann. Inst. H. Poincaré Anal. Non Linéaire 12, 319–337 (1995) · Zbl 0837.46025  Terracini S.: On positive entire solutions to a class of equations with a singular coefficient and critical exponent. Adv. Differential Equations 1, 241–264 (1996) · Zbl 0847.35045  Fieseler K.-H., Tintarev K.: Concentration Compactness: Functional-Analytic Grounds and Applications. Imperial College Press, London (2007) · Zbl 1118.49001
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