×

Multiple interior peak solutions for some singularly perturbed Neumann problems. (English) Zbl 1061.35502

Summary: We consider the problem \(\epsilon^2\Delta u-u+f(u)=0,\) \(u>0\), in \(\Omega, \partial u/\partial\nu=0\) on \(\partial\), where \(\Omega\) is a bounded smooth domain in \(\mathbb R^N, \epsilon>0\) is a small parameter, and \(f\) is a superlinear, subcritical nonlinearity. It is known that this equation possesses boundary spike solutions that concentrate, as \(\epsilon\) approaches zero, at a critical point of the mean curvature function \(H(P),\;P\in\partial\Omega\). It is also proved that this equation has single interior spike solutions at a local maximum point of the distance function \(d(P,\partial\Omega), P\in\Omega\). In this paper, we prove the existence of interior \(K\)-peak \((K\geq2)\) solutions at the local maximum points of the following function: \(\phi(P_1,P_2,\cdots,P_K)=\min_{i,k,l=1, \cdots,K;k\not=l}(d(P_i,\partial\Omega),{1\over2}|P_k-P_l|)\). We first use the Lyapunov-Schmidt reduction method to reduce the problem to a finite-dimensional problem. Then we use a maximizing procedure to obtain multiple interior spikes. The function \(\phi(P_1,\cdots,P_K)\) appears naturally in the asymptotic expansion of the energy functional.

MSC:

35J25 Boundary value problems for second-order elliptic equations
35B25 Singular perturbations in context of PDEs
47J30 Variational methods involving nonlinear operators
47N20 Applications of operator theory to differential and integral equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Agmon, S., Lectures on elliptic boundary value problems, (1965), Van Nostrand Princeton, NJ · Zbl 0151.20203
[2] Adimurthi, G. Mancinne; Yadava, S.L., The role of Mean curvature in a semilinear Neumann problem involving the critical Sobolev exponent, Comm. partial differential equations, 20, 591-631, (1995) · Zbl 0847.35047
[3] Adimurthi, F. Pacella; Yadava, S.L., Interaction between the geometry of the boundary and positive solutions of a semilinear Neumann problem with critical nonlinearity, J. funct. anal., 31, 8-350, (1993) · Zbl 0793.35033
[4] Adimurthi, F. Pacella; Yadava, S.L., Characterization of concentration points and L^{∞}-estimates for solutions involving the critical Sobolev exponent, Differential integral equations, 8, No. 1, 41-68, (1995) · Zbl 0814.35029
[5] Aronson, D.O.; Weinberger, H.F., Multidimensional nonlinear diffusion arising in population genetics, Adv. math., 30, 33-76, (1978) · Zbl 0407.92014
[6] P. Bates and G. Fusco, Equilibria with many nuclei for the Cahn-Hilliard equation, preprint. · Zbl 0990.35016
[7] Bahri, A.; Li, Y., On a MIN-MAX procedure for the existence of a positive solution for certain scalar field equations in R^{N}, Rev. iberoamericana, 6, 1-15, (1990) · Zbl 0731.35036
[8] Cao, D.; Dancer, N.; Noussair, E.; Yan, S., On the existence and profile of multi-peaked solutions to singularly perturbed semilinear Dirichlet problem, Discrete continuous dynam. systems, 2, No. 2, (1996) · Zbl 0947.35073
[9] M. Del Pino, P. Felmer, and J. Wei, Multiple-peak solutions for some singular perturbation problems, Cal. Var. Partial Differential Equations, in press. · Zbl 0974.35041
[10] Jang, J., On spike solutions of singularly perturbed semilinear Dirichlet problems, J. differential equations, 114, 370-395, (1994) · Zbl 0812.35008
[11] E.N. Dancer, A note on asymptotic uniqueness for some nonlinearities which change sign. Rocky Mountain Math. J., in press.
[12] Fleming, W.H.; Souganidis, P.E., Asymptotic series and the method of vanishing viscosity, Indiana univ. math. J., 35, No. 2, 425-447, (1986) · Zbl 0573.35034
[13] 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
[14] Gui, C., Multi-peak solutions for a semilinear Neumann problem, Duke math. J., 84, 739-769, (1996) · Zbl 0866.35039
[15] Gui, C.; Ghoussoub, N., Multi-peak solutions for a semilinear Neumann problem invovling the critical Sobolev exponent, Math. Z., 229, 443-474, (1998) · Zbl 0955.35024
[16] Gidas, B.; Ni, W.-M.; Nirenberg, L., Symmetry of positive solutions of nonlinear elliptic equations in R^{N}, (), 369-402, Part A
[17] Gardner, R.; Peletier, L.A., The set of positive solutions of semilinear equations in large balls, (), 53-72 · Zbl 0625.35030
[18] Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equations of second order, (1983), Springer-Verlag New York · Zbl 0691.35001
[19] Helffer, B.; Sjöstrand, J., Multiple wells in the semi-classical limit, I, Comm. partial differential equations, 9, 337-408, (1984) · Zbl 0546.35053
[20] M. Kowalczyk, Multiple spike layers in the shadow Gierer-Meihardt system: Existence of equilibria and approximate invariant manifold, preprint. · Zbl 0962.35063
[21] Kwong, M.K., Uniqueness of positive solutions of δu −u + u^{p} = 0 in rn, Arch. rational mech. anal., 105, 243-266, (1989) · Zbl 0676.35032
[22] Lin, C.; Ni, W.-M.; Takagi, I., Large amplitude stationary solutions to a chemotaxis systems, J. differential equations, 72, 1-27, (1988) · Zbl 0676.35030
[23] Lions, J.L.; Magenes, E., ()
[24] Li, Y.Y., On a singularly perturbed equation with Neumann boundary condition, Comm. partial differential equations, 23, 487-545, (1998) · Zbl 0898.35004
[25] Ni, W.-M.; Pan, X.; Takagi, I., Singular behavior of least-energy solutions of a semi-linear Neumann problem involving critical Sobolev exponents, Duke math. J., 67, 1-20, (1992) · Zbl 0785.35041
[26] 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
[27] 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
[28] Ni, W.-M.; Takagi, I., Point-condensation generated be a reaction-diffusion system in axially symmetric domains, Japan J. indust. appl. math., 12, 327-365, (1995) · Zbl 0843.35006
[29] 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
[30] Oh, Y.G., Existence of semi-classical bound states of nonlinear Schrödinger equations with potentials of the class (V)_{a}, Comm. partial differential equations, 13, No. 12, 1499-1519, (1988) · Zbl 0702.35228
[31] 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
[32] Pan, X.B., Condensation of least-energy solutions of a semilinear Neumann problem, J. partial differential equations, 8, 1-36, (1995) · Zbl 0814.35039
[33] Pan, X.B., Condensation of least-energy solutions: the effect of boundary conditions, Nonlinear anal., 24, 195-222, (1995) · Zbl 0826.35037
[34] Pan, X.B., Further study on the effect of boundary conditions, J. differential equations, 117, 446-468, (1995) · Zbl 0832.35050
[35] Smoller, J.; Wasserman, A., Global bifurcation of steady-state solutions, J. differential equations, 39, 269-290, (1981) · Zbl 0425.34028
[36] Wang, Z.-Q., On the existence of multiple single-peaked solutions for a semilinear Neumann problem, Arch. rational mech. anal., 120, 375-399, (1992) · Zbl 0784.35035
[37] Ward, M., An asymptotic analysis of localized solutions for some reaction-diffusion models in multidimensional domains, Stud. appl. math., 97, 103-126, (1996) · Zbl 0932.35059
[38] Wei, J., On the construction of single-peaked solutions of a singularly perturbed semilinear Dirichlet problem, J. differential equations, 129, 315-333, (1996) · Zbl 0865.35011
[39] J. Wei, On the effect of the geometry of the domain in a singularly perturbed Dirichlet problem, Differential Integral Equations, in press.
[40] Wei, J., On the interior spike layer solutions of a singularly perturbed Neumann problem, Tohôku math. J., 50, 159-178, (1998) · Zbl 0918.35024
[41] Wei, J., On the boundary spike layer solutions of singularly perturbed semilinear Neumann problem, J. differential equations, 134, (1997)
[42] J. Wei, On the construction of single interior peak solutions for a singularly perturbed Neumann problem, submitted for publication.
[43] Wei, J.; Winter, M., Stationary solutions for the Cahn-Hilliard equation, Ann. inst. H. Poincaré anal. non linéaire, 15, 459-492, (1998) · Zbl 0910.35049
[44] J. Wei and M. Winter, Multiple boundary spike solutions for a wide class of singular perturbation problems, J. London Math. Soc., in press.
[45] Yosida, K., Functional analysis, (1978), Springer-Verlag New York/Berlin · Zbl 0152.32102
[46] Zeidler, E., Nonlinear functional analysis and its applications. I. fixed-point theorems, (1986), Springer-Verlag Berlin/Heidelberg/New York
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.