zbMATH — the first resource for mathematics

On the existence of multiple, single-peaked solutions for a semilinear Neumann problem. (English) Zbl 0784.35035
This paper is concerned with positive solutions of the following semilinear elliptic equation subject to homogeneous Neumann boundary conditions \[ -d\Delta u+u=| u |^{p-2}u \text{ in }\Omega,\quad \partial u/ \partial\nu=0 \text{ on } \partial \Omega \tag{I} \] where \(d\) is a positive constant, \(\Omega\) is a bounded domain in \(\mathbb{R}^ N\) \((N \geq 2)\) with a smooth boundary, and \(\nu\) is the unit outer normal to \(\partial \Omega\). \(p\) satisfies \(2<p<2N/(N-2)\) if \(N \geq 3\), and \(2<p<+\infty\) if \(N=2\).
The goal of this paper is to establish a multiplicity result on the existence of nonconstant positive solutions of (I) and to show how the number of positive solutions is affected by the topology of \(\Omega\) or, more precisely, of \(\partial \Omega\). Moreover, it is proved that all solutions obtained have the property that each solution has at most one local maximum over \(\overline\Omega\), which is achieved at a point on the boundary of \(\Omega\).
G. Mancini and R. Musina (Preprint) have obtained the same existence result. Meanwhile the author has generalized this result to exterior domain problems [see the following review] and to critical exponent problems in bounded domains (Preprint).

35J65 Nonlinear boundary value problems for linear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
Full Text: DOI
[1] Adams, R., Sobolev Spaces, Academic Press, 1975.
[2] Ambrosetti, A. & Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381. · Zbl 0273.49063 · doi:10.1016/0022-1236(73)90051-7
[3] Bahri, A. & Coron, J. M., On a nonlinear elliptic equation involving the Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math. 41 (1988), 253-294. · Zbl 0649.35033 · doi:10.1002/cpa.3160410302
[4] Benci, V. & Cerami, G., The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems, Arch. Rational Mech. Anal. 114 (1991), 79-93. · Zbl 0727.35055 · doi:10.1007/BF00375686
[5] Berestycki, H., Gallou?t, T. & Kavian, O., Equations de champs scalaires euclidiens nonlin?aires dans le plan, C. R. Acad. Sc. Paris, S?rie I Math. 297 (1983), 307-310.
[6] Berestycki, H. & Lions, P.-L., Nonlinear scalar field equations I, existence of a ground state, Arch. Rational Mech. Anal. 82 (1983), 313-375. · Zbl 0533.35029
[7] Coleman, S., Glaser, V. & Martin, A., Action minima among solutions to a class of Euclidean scalar field equations, Comm. Math. Phys. 58 (1978), 211-221. · doi:10.1007/BF01609421
[8] Chow, S. N. & Hale, J. K., Methods of Bifurcation Theory, Springer-Verlag, 1982. · Zbl 0487.47039
[9] Dancer, E. N., The effect of domain shape on the number of positive solutions of certain nonlinear equations, J. Diff. Eqs. 74 (1988), 120-156. · Zbl 0662.34025 · doi:10.1016/0022-0396(88)90021-6
[10] Ding, W.-Y., Positive solutions of ?u + u (N+2)/(N?2) = 0 on contractible domains, J. Part. Diff. Eqs. 2 (1989), 83-88. · Zbl 0694.35067
[11] Ding, W.-Y. & Ni, W.-M., On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Rational Mech. Anal. 91 (1986), 283-308. · Zbl 0616.35029 · doi:10.1007/BF00282336
[12] Fife, P. C., Semilinear elliptic boundary value problems with small parameters, Arch. Rational Mech. Anal. 52 (1973), 205-232. · Zbl 0268.35007 · doi:10.1007/BF00247733
[13] Gierer, A. & Meinhardt, H., A theory of biological pattern formation, Kybernetik 12 (1972), 30-39. · Zbl 0297.92007 · doi:10.1007/BF00289234
[14] Gidas, B., Ni, W.-M. & Nirenberg, L., Symmetry of positive solutions of nonlinear elliptic equations in R n , Advances in Math., Supplementary Studies 7A (1981), 369-402. · Zbl 0469.35052
[15] Gilbarg, D. & Trudinger, N. S., Elliptic Partial Differential Equations of Second Order, Second Edition, Springer-Verlag, 1983. · Zbl 0562.35001
[16] Keller, E. F. & Segel, L. A., Initiation of slime model aggregation viewed as an instability, J. Theor. Biol. 26 (1970), 399-415. · Zbl 1170.92306 · doi:10.1016/0022-5193(70)90092-5
[17] Kwong, M. K., Uniqueness of positive solutions ?u ? u + u p = 0 in R n , Arch. Rational Mech. Anal. 105 (1989), 243-266. · Zbl 0676.35032 · doi:10.1007/BF00251502
[18] Lions, P. L., The concentration-compactness principle in the calculus of variations. The locally compact case, Part 1 and Part 2, Ann. Inst. H. Poincar?, Anal. Nonlin?aire 1 (1984), 109-145, 223-283.
[19] Lin, C.-S. & Ni, W.-M., On the diffusion coefficient of a semilinear Neumann problem, in Calculus of Variations and Partial Differential Equations, (S. Hildebrandt, D. Kinderlehrer & M. Miranda, Eds.), 160-174, Lecture Notes in Math. 1340, Springer-Verlag, 1988.
[20] Lin, C.-S., Ni, W.-M. & Takagi, I., Large amplitude stationary solutions to a chemotaxis system, J. Diff, Eqs. 72 (1988), 1-27. · Zbl 0676.35030 · doi:10.1016/0022-0396(88)90147-7
[21] McLeod, K. & Serrin, J., Uniqueness of positive radial solutions of ?u + f(u) = 0 in R n , Arch. Rational Mech. Anal. 99 (1987), 115-145. · Zbl 0667.35023 · doi:10.1007/BF00275874
[22] Ni, W.-M. & Takagi, I., On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math. 45 (1991), 819-851. · Zbl 0754.35042 · doi:10.1002/cpa.3160440705
[23] Takagi, I., Point-condensation for a reaction-diffusion system, J. Diff. Eqs. 61 (1986), 208-249. · Zbl 0627.35049 · doi:10.1016/0022-0396(86)90119-1
[24] Wang, Z.-Q., On the existence of positive solutions for semilinear Neumann problems in exterior domains, to appear in Comm. Partial Diff. Eqs.
[25] Wang, Z.-Q., The effect of the domain geometry on the number of positive solutions of Neumann problems with critical exponents, preprint.
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.