×

zbMATH — the first resource for mathematics

Singular behavior of least-energy solutions of a semilinear Neumann problem involving critical Sobolev exponents. (English) Zbl 0785.35041
We study the behavior of least-energy solutions, as \(d\to 0\), of the singularly perturbed semilinear Neumann problem \[ d\Delta u-u+u^ \tau=0\text{ in }\Omega,\;u>0\text{ in }\Omega,\;{\partial u\over\partial\nu}=0\text{ on }\partial\Omega, \tag{1.1} \] where \(\Delta=\sum^ n_{j=1}\partial^ 2/\partial x^ 2_ j\) is the Laplace operator, \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^ n\), \(n\geq 3\), \(v\) is the unit outer normal to \(\partial\Omega\), \(\tau=(n+2)/(n-2)\), and \(d>0\) is a constant. By a least-energy solution of (1.1) we mean a (classical) solution of (1.1) which minimizes the “energy” functional \[ J_ d(u)=\int\left\{{1\over 2}(d|\nabla u|^ 2+u^ 2)-{1\over\tau+1}u_ +^{\tau+1}\right\}, \] where \(u_ +=\max(u,0)\), among all the solutions of (1.1). The purpose of this paper is to investigate the behavior of the least-energy solution \(u_ d\) of (1.1) as \(d\to 0\). We establish that \(u_ d\to 0\) in \(\Omega\) as \(d\to 0\) and the (global) maximum of \(u_ d\) in \(\overline\Omega\) is assumed at exactly one point \(P_ d\) which must lie on the boundary \(\partial\Omega\). However, in contrast to the case of \(\tau<(n+2)/(n-2)\) we now have \(\| u_ d\|_{L^ \infty(\Omega)}\to\infty\) as \(d\to 0\). Moreover, after a change of variables and a suitable rescaling, the function \(u_ d/\| u_ d\|_{L^ \infty(\Omega)}\) converges in a small neighborhood of \(P_ d\) to a positive solution of \(\Delta U+U^ \tau=0\) in \(\mathbb{R}^ n\).

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B40 Asymptotic behavior of solutions to PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Adimurthi and G. Mancini, The Neumann problem for elliptic equations with critical non-linearity , · Zbl 0836.35048
[2] Adimurthi and S. L. Yadava, On a conjecture of Lin-Ni for the semilinear Neumann problem , · Zbl 0787.35030
[3] Adimurthi and S. L. Yadava, Existence and nonexistence of positive radial solutions for the Sobolev critical exponent problem with Neumann boundary conditions , · Zbl 0839.35041 · doi:10.1007/BF00380771
[4] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications , J. Functional Analysis 14 (1973), 349-381. · Zbl 0273.49063 · doi:10.1016/0022-1236(73)90051-7
[5] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents , Comm. Pure Appl. Math. 36 (1983), no. 4, 437-477. · Zbl 0541.35029 · doi:10.1002/cpa.3160360405
[6] C. Budd, M. C. Knaap, and L. A. Peletier, Asymptotic behavior of solutions of elliptic equations with critical exponents and Neumann boundary conditions , · Zbl 0733.35038 · doi:10.1017/S0308210500024707
[7] L. A. Caffarelli, B. Gidas, and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth , Comm. Pure Appl. Math. 42 (1989), no. 3, 271-297. · Zbl 0702.35085 · doi:10.1002/cpa.3160420304
[8] W. X. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations , Duke Math. J. 63 (1991), no. 3, 615-622. · Zbl 0768.35025 · doi:10.1215/S0012-7094-91-06325-8
[9] M. Comte and M. C. Knaap, Solutions of elliptic equations involving critical Sobolev exponents with Neumann boundary conditions , Manuscripta Math. 69 (1990), no. 1, 43-70. · Zbl 0717.35029 · doi:10.1007/BF02567912 · eudml:155545
[10] M. Comte and M. C. Knaap, Existence of solutions of elliptic equations involving critical Sobolev exponents with Neumann boundary conditions in general domains , · Zbl 0868.35036 · doi:10.1017/S0308210500030596
[11] W.-Y. Ding and W.-M. Ni, On the existence of positive entire solutions of a semilinear elliptic equation , Arch. Rational Mech. Anal. 91 (1986), no. 4, 283-308. · Zbl 0616.35029 · doi:10.1007/BF00282336
[12] H. Egnell, Asymptotic results for finite-energy solutions of semilinear elliptic equations , to appear in J. Differential Equations. · Zbl 0778.35009 · doi:10.1016/0022-0396(92)90103-T
[13] B. Gidas, W. M. Ni, and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in \(\mathbf R\spn\) , Mathematical analysis and applications, Part A, Adv. in Math. Suppl. Stud., vol. 7, Academic Press, New York, 1981, pp. 369-402. · Zbl 0469.35052
[14] C. S. Lin and W.-M. Ni, On the diffusion coefficient of a semilinear Neumann problem , Calculus of variations and partial differential equations (Trento, 1986) eds. S. Hildebrandt, D. Kinderlehrer, and M. Miranda, Lecture Notes in Math., vol. 1340, Springer, Berlin, 1988, pp. 160-174. · Zbl 0704.35050 · doi:10.1007/BFb0082894
[15] C.-S. Lin, W.-M. Ni, and I. Takagi, Large amplitude stationary solutions to a chemotaxis system , J. Differential Equations 72 (1988), no. 1, 1-27. · Zbl 0676.35030 · doi:10.1016/0022-0396(88)90147-7
[16] W.-M. Ni, Recent progress in semilinear elliptic equations , RIMS Kokyuroku 679 (1989), 1-39.
[17] W. M. Ni, On the positive radial solutions of some semilinear elliptic equations on \(\mathbf R\spn\) , Appl. Math. Optim. 9 (1983), no. 4, 373-380. · Zbl 0527.35026 · doi:10.1007/BF01460131
[18] W.-M. Ni, Some aspects of semilinear elliptic equations on \(\mathbf R^ n\) , Nonlinear diffusion equations and their equilibrium states, II (Berkeley, CA, 1986) eds. W.-M. Ni, L. A. Peletier, and J. Serrin, Math. Sci. Res. Inst. Publ., vol. 13, Springer, New York, 1988, pp. 171-205. · Zbl 0676.35026
[19] W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem , Comm. Pure Appl. Math. 44 (1991), no. 7, 819-851. · Zbl 0754.35042 · doi:10.1002/cpa.3160440705
[20] I. Takagi, Point-condensation for a reaction-diffusion system , J. Differential Equations 61 (1986), no. 2, 208-249. · Zbl 0627.35049 · doi:10.1016/0022-0396(86)90119-1
[21] N. S. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds , Ann. Scuola Norm. Sup. Pisa (3) 22 (1968), 265-274. · Zbl 0159.23801 · numdam:ASNSP_1968_3_22_2_265_0 · eudml:83458
[22] X.-J. Wang, Neumann problems of semilinear elliptic equations involving critical Sobolev exponents , J. Differential Equations 93 (1991), no. 2, 283-310. · Zbl 0766.35017 · doi:10.1016/0022-0396(91)90014-Z
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.