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Multipeak solutions for a semilinear Neumann problem. (English) Zbl 0866.35039
The paper is concerned with the semilinear Neumann problem: $\varepsilon^2 \Delta u-u+f(u) =0,\;u>0, \text{ in } \Omega,\quad {\partial u\over \partial\nu} =0\text{ on } \partial \Omega,$ where $$\Omega$$ is a bounded domain in $$\mathbb{R}^N$$, $$\nu$$ is the outer normal to $$\partial\Omega$$ and $$\varepsilon$$ is a positive constant. In addition to suitable conditions on $$f(t)$$, typically satisfied by the function $$f(t)= t^p-at^q$$ if $$a\geq 0$$ and $$1<q<p <(N+2)/(N-2)$$, the domain $$\Omega$$ is assumed to satisfy the condition that there exist $$k$$ disjoint patches $$\Lambda_1$$, $$\Lambda_2, \dots, \Lambda_k$$ on $$\partial \Omega$$ such that $$\max_{P\in \Lambda_i} H(P)> \max_{P\in\partial \Lambda_i} H(P)$$, where $$H(P)$$ denotes the mean curvature of $$\partial \Omega$$ at $$P$$. Under these conditions, the author proves the existence of a classical solution $$u_\varepsilon$$ which has exactly $$k$$ local maxima, precisely one on each $$\Lambda_i (i=1,2, \dots,k)$$, and then analyses the asymptotic behavior as $$\varepsilon \downarrow 0$$.

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35J20 Variational methods for second-order elliptic equations 35B25 Singular perturbations in context of PDEs
##### Keywords:
local maxima of a solution; asymptotic behavior
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##### References:
  S. Alama and Y. Y. Li, On “multibump” bound states for certain semilinear elliptic equations , Indiana Univ. Math. J. 41 (1992), no. 4, 983-1026. · Zbl 0796.35043 · doi:10.1512/iumj.1992.41.41052  H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state , Arch. Rational Mech. Anal. 82 (1983), no. 4, 313-345. · Zbl 0533.35029  H. Brezis and T. Kato, Remarks on the Schrödinger operator with singular complex potentials , J. Math. Pures Appl. (9) 58 (1979), no. 2, 137-151. · Zbl 0408.35025  C.-C. Chen and C.-S. Lin, Uniqueness of the ground state solutions of $$\Delta u+f(u)=0$$ in $$\mathbf R^ n,\;n\geq 3$$ , Comm. Partial Differential Equations 16 (1991), no. 8-9, 1549-1572. · Zbl 0753.35034 · doi:10.1080/03605309108820811  V. Coti Zelati and P. H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials , J. Amer. Math. Soc. 4 (1991), no. 4, 693-727. JSTOR: · Zbl 0744.34045 · doi:10.2307/2939286 · links.jstor.org  V. Coti Zelati and P. H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on $$\mathbf R^ n$$ , Comm. Pure Appl. Math. 45 (1992), no. 10, 1217-1269. · Zbl 0785.35029 · doi:10.1002/cpa.3160451002  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  D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order , 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. · Zbl 0562.35001  C. Gui, Existence of multibump solutions for nonlinear Schrödinger equations via variational method , to appear in Comm. Partial Differential Equations. · Zbl 0857.35116 · doi:10.1080/03605309608821208  M. K. Kwong and L. Zhang, Uniqueness of the positive solution of $$\Delta u+f(u)=0$$ in an annulus , Differential Integral Equations 4 (1991), no. 3, 583-599. · Zbl 0724.34023  Y. Y. Li, On $$-\Delta u=K(x)u^ 5$$ in $$\mathbf R^ 3$$ , Comm. Pure Appl. Math. 46 (1993), no. 3, 303-340. · Zbl 0799.35068 · doi:10.1002/cpa.3160460302  Y. Y. Li, Prescribing scalar curvature on $$S^ 3,\;S^ 4$$ and related problems , J. Funct. Anal. 118 (1993), no. 1, 43-118. · Zbl 0790.53040 · doi:10.1006/jfan.1993.1138  Y. Y. Li, Prescribing scalar curvature on $$S^ n$$ and related problems. I , J. Differential Equations 120 (1995), no. 2, 319-410. · Zbl 0827.53039 · doi:10.1006/jdeq.1995.1115  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  1 P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I , Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 2, 109-145. · Zbl 0541.49009 · numdam:AIHPC_1984__1_2_109_0 · eudml:78069  2 P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II , Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 4, 223-283. · Zbl 0704.49004 · numdam:AIHPC_1984__1_4_223_0 · eudml:78074  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  W.-M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem , Duke Math. J. 70 (1993), no. 2, 247-281. · Zbl 0796.35056 · doi:10.1215/S0012-7094-93-07004-4  E. Sere, Existence of infinitely many homoclinic orbits in Hamiltonian systems , Math. Z. 209 (1992), no. 1, 27-42. · Zbl 0725.58017 · doi:10.1007/BF02570817 · eudml:174347
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