zbMATH — the first resource for mathematics

A concentration phenomenon for semilinear elliptic equations. (English) Zbl 1266.35074
In the very interesting paper under review the authors consider the equation \[ -\Delta u+V(x)u=Q_n(x)|{u}|^{p-2}u \] in a domain \(\Omega \subset \mathbb R^N\) with zero Dirichlet boundary conditions and where \(p\in(2,2^\ast).\) It is assumed that \(V\geqq 0\) and \(Q_n\) are bounded functions which are positive in a sub-region of \(\Omega\) and negative outside, and such that the sets \(\{Q_n>0\}\) shrink to a point \(x_0\in\Omega\) as \(n\to \infty.\) The main result of the paper states that if \(u_n\) is a nontrivial solution corresponding to \(Q_n,\) then the sequence \((u_n)\) concentrates at \(x_0\) with respect to the \(H^1\) and certain \(L^q\)-norms. It is also shown that if the sets \(\{Q_n>0\}\) shrink to two points and \(u_n\) are ground state solutions, then they concentrate at one of these points.

35J61 Semilinear elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
Full Text: DOI arXiv
[1] Ambrosetti, A., Arcoya, D., Gámez, J.L.: Asymmetric bound states of differential equations in nonlinear optics. Rend. Sem. Math. Univ. Padova 100, 231–247 (1998). http://www.numdam.org/item?id=RSMUP_1998__100__231_0 · Zbl 0922.34020
[2] Bandle, C., Marcus, M.: ”Large” solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behaviour. J. Anal. Math. 58, 9–24 (1992). doi: 10.1007/BF02790355 (Festschrift on the occasion of the 70th birthday of Shmuel Agmon) · Zbl 0802.35038
[3] Berestycki H., Capuzzo-Dolcetta I., Nirenberg L.: Variational methods for indefinite superlinear homogeneous elliptic problems. NoDEA Nonlinear Differ. Equ. Appl. 2(4), 553–572 (1995) · Zbl 0840.35035
[4] Bonheure D., Gomes J.M., Habets P.: Multiple positive solutions of superlinear elliptic problems with sign-changing weight. J. Differ. Equ. 214(1), 36–64 (2005). doi: 10.1016/j.jde.2004.08.009 · Zbl 1210.35089
[5] Brézis H., Véron L.: Removable singularities for some nonlinear elliptic equations. Arch. Ration. Mech. Anal. 75(1), 1–6 (1980). doi: 10.1007/BF00284616 · Zbl 0459.35032
[6] Buryak A.V., Trapani P.D., Skryabin D.V., Trillo S.: Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications. Phys. Rep. 370(2), 63–235 (2002). doi: 10.1016/S0370-1573(02)00196-5 · Zbl 0998.78009
[7] Costa D.G., Tehrani H.: Existence of positive solutions for a class of indefinite elliptic problems in $${\(\backslash\)mathbb{R}\^{N}}$$ . Calc. Var. Partial Differ. Equ. 13(2), 159–189 (2001) · Zbl 1077.35045
[8] Dror, N., Malomed, B.A.: Solitons supported by localized nonlinearities in periodic media. Phys. Rev. A 83, 033,828 (2011). doi: 10.1103/PhysRevA.83.033828
[9] Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Grundlehren der Mathematischen Wissenschaften. Fundamental Principles of Mathematical Sciences, Vol. 224, 2nd edn. Springer, Berlin, 1983 · Zbl 0562.35001
[10] Girão P.M., Gomes J.M.: Multibump nodal solutions for an indefinite superlinear elliptic problem. J. Differ. Equ. 247(4), 1001–1012 (2009). doi: 10.1016/j.jde.2009.04.018 · Zbl 1173.35060
[11] Kartashov Y.V., Malomed B.A., Torner L.: Solitons in nonlinear lattices. Rev. Mod. Phys. 83, 247–305 (2011). doi: 10.1103/RevModPhys.83.247
[12] López-Gómez J.: Varying bifurcation diagrams of positive solutions for a class of indefinite superlinear boundary value problems. Trans. Am. Math. Soc. 352(4), 1825–1858 (2000). doi: 10.1090/S0002-9947-99-02352-1 · Zbl 0940.35095
[13] Pendry J.B., Schurig D., Smith D.R.: Controlling electromagnetic fields. Science 312(5781), 1780–1782 (2006). doi: 10.1126/science.1125907 · Zbl 1226.78003
[14] Shalaev V.M.: Optical negative-index metamaterials. Nat. Photon. 1(1), 41–48 (2007). doi: 10.1038/nphoton.2006.49
[15] Smith D.R., Pendry J.B., Wiltshire M.C.K.: Metamaterials and negative refractive index. Science 305(5685), 788–792 (2004). doi: 10.1126/science.1096796
[16] Strauss, W.A.: The nonlinear Schrödinger equation. Contemporary developments in continuum mechanics and partial differential equations. Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977. North-Holland Math. Stud., Vol. 30. North-Holland, Amsterdam, 452–465, 1978
[17] Stuart C.A.: Bifurcation in L p (R N ) for a semilinear elliptic equation. Proc. London Math. Soc. (3) 57(3), 511–541 (1988). doi: 10.1112/plms/s3-57.3.511 · Zbl 0673.35005
[18] Stuart C.A.: Self-trapping of an electromagnetic field and bifurcation from the essential spectrum. Arch. Ration. Mech. Anal. 113(1), 65–96 (1991). doi: 10.1007/BF00380816 · Zbl 0745.35044
[19] Stuart C.A.: Guidance properties of nonlinear planar waveguides. Arch. Ration. Mech. Anal. 125(2), 145–200 (1993). doi: 10.1007/BF00376812 · Zbl 0801.35136
[20] Stuart C.A.: Existence and stability of TE modes in a stratified non-linear dielectric. IMA J. Appl. Math. 72(5), 659–679 (2007). doi: 10.1093/imamat/hxm033 · Zbl 1138.78316
[21] Veselago, V.G.: The electrodynamics of substances with simultaneously negative values of {\(\epsilon\)} and {\(\mu\)}. Physics-Uspekhi 10(4), 509–514 (1968). doi: 10.1070/PU1968v010n04ABEH003699 . http://ufn.ru/en/articles/1968/4/e/
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.