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Concentration of solutions for some singularly perturbed mixed problems: existence results. (English) Zbl 1214.35024
Let \(\Omega\) be a bounded open subset of \(\mathbb R^n\) with smooth boundary. Let \(p\in \,]1,\frac{n+2}{n-2}[\). Let \(\nu\) be the outward unit normal to \(\partial\Omega\). Let \(\partial_{\mathcal D}\Omega\), \(\partial_{\mathcal N}\Omega\) be disjoint open subsets of the boundary \(\partial\Omega\) of \(\Omega\) such that the union of the closures of \(\partial_{\mathcal D}\Omega\) and of \(\partial_{\mathcal N}\Omega\) equals \(\partial\Omega\).
The authors show that under suitable assumptions on the geometry of \(\partial\Omega\), if \(\overline{Q}\) belongs to the intersection of the closures of \(\partial_{\mathcal D}\Omega\) and of \(\partial_{\mathcal N}\Omega\), then the mixed boundary value problem
\[ -\varepsilon^2\Delta u + u=u^p\quad\text{in }\Omega, \qquad u>0 \quad\text{on }\Omega, \qquad \frac{\partial u}{\partial \nu}=0\quad\text{on }\partial_{\mathcal N}\Omega, \qquad u=0\quad\text{on }\partial_{\mathcal D}\Omega, \] admits a family of solutions \(u_\varepsilon\) which concentrate at \(\overline{Q}\) as the parameter \(\varepsilon>0\) shrinks to \(0\).

MSC:
35J61 Semilinear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35B25 Singular perturbations in context of PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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[1] Ambrosetti A., Badiale M., Cingolani S.: Semiclassical states of nonlinear Schrödinger equations. Arch. Ration. Mech. Anal. 140, 285–300 (1997) · Zbl 0896.35042 · doi:10.1007/s002050050067
[2] Ambrosetti, A., Malchiodi, A.: Perturbation Methods and Semilinear Elliptic Problems on \({\mathbb{R}^n}\) . Birkhäuser, Progr. in Math., Vol. 240, 2005 · Zbl 1115.35004
[3] Ambrosetti A., Malchiodi A.: Nonlinear Analysis and Semilinear Elliptic Problems. Cambridge Studies in Advanced Mathematics, Vol. 104. Cambridge University Press, Cambridge (2007) · Zbl 1125.47052
[4] Ambrosetti A., Malchiodi A., Ni W.M.: Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres, Part I. Comm. Math. Phys. 235, 427–466 (2003) · Zbl 1072.35019 · doi:10.1007/s00220-003-0811-y
[5] Ambrosetti A., Malchiodi A., Ni W.M.: Singularly Perturbed Elliptic Equations with Symmetry: existence of Solutions Concentrating on Spheres, Part II. Indiana Univ. Math. J. 53(2), 297–329 (2004) · Zbl 1081.35008 · doi:10.1512/iumj.2004.53.2400
[6] Ambrosetti A., Malchiodi A., Secchi S.: Multiplicity results for some nonlinear singularly perturbed elliptic problems on \({\mathbb{R}^n}\) . Arch. Ration. Mech. Anal. 159(3), 253–271 (2001) · Zbl 1040.35107 · doi:10.1007/s002050100152
[7] Ambrosetti A., Rabinowitz P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973) · Zbl 0273.49063 · doi:10.1016/0022-1236(73)90051-7
[8] Badiale, M., D’Aprile, T.: Concentration around a sphere for a singularly perturbed Schrödinger equation. Nonlinear Anal. 49(7), Ser. A: Theory Methods, 947–985 (2002) · Zbl 1018.35021
[9] Bartsch, T., Peng, S.: Semiclassical symmetric Schrödinger equations: existence of solutions concentrating simultaneously on several spheres, preprint · Zbl 1133.35087
[10] Benci V., D’Aprile T.: The semiclassical limit of the nonlinear Schrödinger equation in a radial potential. J. Differ. Equ. 184(1), 109–138 (2002) · Zbl 1060.35129 · doi:10.1006/jdeq.2001.4138
[11] Casten R.G., Holland C.J.: Instability results for reaction diffusion equations with Neumann boundary conditions. J. Differ. Eq. 27(2), 266–273 (1978) · Zbl 0359.35039 · doi:10.1016/0022-0396(78)90033-5
[12] Dancer E.N., Yan S.: Multipeak solutions for a singularly perturbed Neumann problem. Pac. J. Math. 189(2), 241–262 (1999) · Zbl 0933.35070 · doi:10.2140/pjm.1999.189.241
[13] D’Aprile T.: On a class of solutions with non-vanishing angular momentum for nonlinear Schrödinger equations. Diff. Int. Equ. 16(3), 349–384 (2003) · Zbl 1031.35130
[14] Del Pino M., Felmer P., Kowalczyk M.: Boundary spikes in the Gierer–Meinhardt system. Commun. Pure Appl. Anal. 1(4), 437–456 (2002) · Zbl 1163.35354 · doi:10.3934/cpaa.2002.1.437
[15] Del Pino M., Felmer P., Wei J.: On the role of the mean curvature in some singularly perturbed Neumann problems. S.I.A.M. J. Math. Anal. 31, 63–79 (1999) · Zbl 0942.35058
[16] Del Pino M., Kowalczyk M., Wei J.: Multi-bump ground states of the Gierer– Meinhardt system in \({\mathbb{R}^2}\) . Ann. Inst. H. Poincaré Anal. Non Linéaire 20(1), 53–85 (2003) · Zbl 1114.35065 · doi:10.1016/S0294-1449(02)00024-0
[17] Floer A., Weinstein A.: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 69, 397–408 (1986) · Zbl 0613.35076 · doi:10.1016/0022-1236(86)90096-0
[18] Garcia Azorero, J., Malchiodi, A., Montoro, L., Peral, I.: Concentration of Solutions for Some Singularly Perturbed Mixed Problems: Asymptotics of Minimal Energy Solutions, to appear in Annales de l’Institut Henri Poincaré Analyse Non Linéaire · Zbl 1194.35037
[19] Gierer, A., Meinhardt, H.: A theory of biological pattern formation. Kybernetik (Berlin), 12, 30–39 (1972) · Zbl 0297.92007 · doi:10.1007/BF00289234
[20] Grisvard, P.: Elliptic problems in nonsmooth domains. Monographs and Studies in Mathematics, 24. Pitman (Advanced Publishing Program), Boston, MA, 1985 · Zbl 0695.35060
[21] Grossi M., Pistoia A., Wei J.: Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory. Calc. Var. Partial Differ. Equ. 11(2), 143–175 (2000) · Zbl 0964.35047 · doi:10.1007/PL00009907
[22] Gui C.: Multipeak solutions for a semilinear Neumann problem. Duke Math. J. 84, 739–769 (1996) · Zbl 0866.35039 · doi:10.1215/S0012-7094-96-08423-9
[23] Gui C., Wei J.: On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems. Can. J. Math. 52(3), 522–538 (2000) · Zbl 0949.35052 · doi:10.4153/CJM-2000-024-x
[24] Gui C., Wei J., Winter M.: Multiple boundary peak solutions for some singularly perturbed Neumann problems. Ann. Inst. H. Poincaré Anal. Non Linéaire 17(1), 47–82 (2000) · Zbl 0944.35020 · doi:10.1016/S0294-1449(99)00104-3
[25] Kwong M.K.: Uniqueness of positive solutions of \({-\Delta{u} + u + u^p = 0}\) in \({\mathbb{R}^n}\) . Arch. Ration. Mech. Anal. 105, 243–266 (1989) · Zbl 0676.35032 · doi:10.1007/BF00251502
[26] Li Y.Y.: On a singularly perturbed equation with Neumann boundary conditions. Comm. Partial Differ. Equ. 23, 487–545 (1998) · Zbl 0898.35004
[27] Li Y.Y., Nirenberg L.: The Dirichlet problem for singularly perturbed elliptic equations. Comm. Pure Appl. Math. 51, 1445–1490 (1998) · Zbl 0933.35083 · doi:10.1002/(SICI)1097-0312(199811/12)51:11/12<1445::AID-CPA9>3.0.CO;2-Z
[28] Lin C.S., Ni W.M., Takagi I.: Large amplitude stationary solutions to a chemotaxis systems. J. Differ. Equ. 72, 1–27 (1988) · Zbl 0676.35030 · doi:10.1016/0022-0396(88)90147-7
[29] Mahmoudi F., Malchiodi A.: Concentration on minimal submanifolds for a singularly perturbed Neumann problem. Adv. Math. 209460–525 (2007) · Zbl 1160.35011 · doi:10.1016/j.aim.2006.05.014
[30] Malchiodi A.: Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains.. G.A.F.A. 151162–1222 (2005) · Zbl 1087.35010
[31] Malchiodi A., Montenegro M.: Boundary concentration phenomena for a singularly perturbed elliptic problem. Comm. Pure Appl. Math 15, 1507–1568 (2002) · Zbl 1124.35305 · doi:10.1002/cpa.10049
[32] Malchiodi A., Montenegro M.: Multidimensional boundary-layers for a singularly perturbed Neumann problem. Duke Math. J. 124105–143 (2004) · Zbl 1065.35037 · doi:10.1215/S0012-7094-04-12414-5
[33] Malchiodi A., Ni W.M., Wei J.: Multiple clustered layer solutions for semilinear Neumann problems on a ball. Ann. Inst. H. Poincaré Anal. Non Linéaire 22, 143–163 (2005) · Zbl 1207.35141 · doi:10.1016/j.anihpc.2004.05.003
[34] Malchiodi A., Wei J.: On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems. Comm. Pure Appl. Math. 48731–768 (1995) · Zbl 0838.35009 · doi:10.1002/cpa.3160480704
[35] Matano H.: Asymptotic behavior and stability of solutions of semilinear diffusion equations. Publ. Res. Inst. Math. Sci. 15401–454 (1979) · Zbl 0445.35063 · doi:10.2977/prims/1195188180
[36] Molle R., Passaseo D.: Concentration phenomena for solutions of superlinear elliptic problems. Ann. Inst. H. Poincaré Anal. Non Linéaire 2363–84 (2006) · Zbl 1293.35114 · doi:10.1016/j.anihpc.2005.02.002
[37] Ni W.M.: Diffusion, cross-diffusion, and their spike-layer steady states. Notices Am. Math. Soc. 459–18 (1998) · Zbl 0917.35047
[38] Ni W.M., Takagi I.: On the shape of least-energy solution to a semilinear Neumann problem. Comm. Pure Appl. Math. 41, 819–851 (1991) · Zbl 0754.35042 · doi:10.1002/cpa.3160440705
[39] Ni W.M., Takagi I.: Locating the peaks of least-energy solutions to a semilinear Neumann problem. Duke Math. J. 70, 247–281 (1993) · Zbl 0796.35056 · doi:10.1215/S0012-7094-93-07004-4
[40] Ni W.M., Wei J.: On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems. Comm. Pure Appl. Math. 48, 731–768 (1995) · Zbl 0838.35009 · doi:10.1002/cpa.3160480704
[41] Oh Y.G.: On positive Multi-lump bound states of nonlinear Schrödinger equations under multiple well potentials. Comm. Math. Phys. 131, 223–253 (1990) · Zbl 0753.35097 · doi:10.1007/BF02161413
[42] Stein E.: Singular integrals and differentiability of functions. Princeton University Press, Princeton (1970) · Zbl 0207.13501
[43] Turing A.M.: The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. Ser B Biol. Sci. 237, 37–72 (1952) · Zbl 1403.92034 · doi:10.1098/rstb.1952.0012
[44] Wang Z.Q.: On the existence of multiple, single-peaked solutions for a semilinear Neumann problem. Arch. Ration. Mech. Anal. 120, 375–399 (1992) · Zbl 0784.35035 · doi:10.1007/BF00380322
[45] Wei J.: On the boundary spike layer solutions of a singularly perturbed semilinear Neumann problem. J. Differ. Equ. 134, 104–133 (1997) · Zbl 0873.35007 · doi:10.1006/jdeq.1996.3218
[46] Wei J.: On the effect of domain geometry in singular perturbation problems. Differ. Int. Equ. 13(1-3), 15–45 (2000) · Zbl 0970.35034
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