×

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
PDF BibTeX XML Cite
Full Text: DOI arXiv Link

References:

[1] L. Abatangelo and S. Terracini, A note on the complete rotational invariance of biradial solutions to semilinear elliptic equations. Preprint, 2009. · Zbl 1218.35105
[2] 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
[3] 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
[4] 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
[5] 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
[6] 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
[7] 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
[8] Cingolani S.: Semiclassical stationary states of nonlinear Schrödinger equations with an external magnetic field. J. Differential Equations 188, 52–79 (2003) · Zbl 1062.81056
[9] Cingolani S., Clapp M.: Intertwining semiclassical bound states to a nonlinear magnetic Schrödinger equation. Nonlinearity 22, 2309–2331 (2009) · Zbl 1173.35678
[10] Cingolani S., Secchi S.: Semiclassical limit for nonlinear Schrödinger equations with electromagnetic fields. J. Math. Anal. Appl. 275, 108–130 (2002) · Zbl 1014.35087
[11] Cingolani S., Secchi S.: Semiclassical states for NLS equations with magnetic potentials having polynomial growths. J. Math. Phys. 46, 053503 (2005) · Zbl 1110.81081
[12] 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
[13] 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
[14] L. C. Evans, Partial differential equations. 2nd ed., Grad. Stud. Math. 19, American Mathematical Society, Providence, RI, 2010. · Zbl 1194.35001
[15] 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.
[16] 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
[17] 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
[18] 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
[19] 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
[20] 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
[21] Leinfelder H.: Gauge invariance of Schrödinger operators and related spectral properties. J. Operator Theory 9, 163–179 (1983) · Zbl 0528.35024
[22] 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
[23] 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
[24] Reed M., Simon B.: Methods of modern mathematical physics. II. Fourier analysis, self-adjointness. Academic Press, New York (1975) · Zbl 0308.47002
[25] Schindler I., Tintarev K.: A nonlinear Schrödinger equation with external magnetic field. Rostock. Math. Kolloq. 56, 49–54 (2002) · Zbl 1140.35567
[26] 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
[27] 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
[28] Fieseler K.-H., Tintarev K.: Concentration Compactness: Functional-Analytic Grounds and Applications. Imperial College Press, London (2007) · Zbl 1118.49001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.