zbMATH — the first resource for mathematics

Existence and uniqueness results on single-peaked solutions of a semilinear problem. (English) Zbl 0905.35033
The paper is concerned with existence, multiplicity and the shape of positive solutions of the Dirichlet problem \[ -\varepsilon^2 \Delta u+u =Q(x)u^{p-1} \quad\text{in } \Omega, \quad u|_{\partial \Omega} =0. \] Here \(\Omega\subset \mathbb{R}^n\), \(n\geq 3\), is a bounded smooth domain, \(Q\) a positive smooth function, \(\varepsilon\) a (small) parameter and \(2<p<2n/(n-2)\). The authors investigate a correspondence between the number of critical points of the coefficient \(Q\) and the number of single-peaked solutions. A positive solution is called single-peaked if it has precisely one maximum point.
Under the assumption that \(Q\) has exactly \(k\) critical points, which are all nondegenerate, it is shown that, for \(\varepsilon\) small enough, there are precisely \(k\) distinct single-peaked solutions. The maximum points of the respective solutions converge to the critical points of \(Q\), as \(\varepsilon \searrow 0\). These results hold irrespective of the geometry and topology of the domain \(\Omega\).
The more delicate case of isolated degenerate critical points of \(Q\) is also treated. Here the authors have an analogous existence result, but the precise number (“uniqueness”) of single-peaked solutions remains open.

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
35J20 Variational methods for second-order elliptic equations
Full Text: DOI Numdam EuDML
[1] Bahri, A.; Coron, J.M., On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. pure. appl. math., Vol. 41, 253-294, (1988) · Zbl 0649.35033
[2] Bahri, A.; Li, Y.Y.; Rey, O., On a variational problem with lack of compactness: the topological effect of the critical points at infinity, Cal. var., Vol. 3, 67-93, (1995) · Zbl 0814.35032
[3] 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, 79-93, (1994) · Zbl 0727.35055
[4] \scD. Cao, E. Noussair and \scS. Yan, Solutions with multiple “Peaks” for nonlinear elliptic equations, preprint.
[5] Dancer, E.N., The effect of the domain shape on the number of positive solutions of certain equations, J. diff. equs., Vol. 74, 316-339, (1988) · Zbl 0729.35050
[6] Dancer, E.N., The effect of the domain shape on the number of positive solutions of certain nonlinear equations, II, J. diff. equa., Vol. 87, 316-339, (1990) · Zbl 0729.35050
[7] Gidas, B.; Ni, Wei-Ming; Nirenberg, L., Symmetry and related properties via the maximum principle, Comm. math. phys., Vol. 68, 209-243, (1979) · Zbl 0425.35020
[8] Glangetas, L., Uniqueness of positive solutions of a nonlinear elliptic equation involving the critical exponent, Nonlinear anal. T.M.A., Vol. 20, No. 5, 571-603, (1993) · Zbl 0797.35048
[9] \scW. M. Ni and \scJ. Wei, On the location and profile of spike-layer solutions to singularity perturbed semilinear problems, to appear in Comm. Pure Appl. Math. · Zbl 0838.35009
[10] Ni, W.M.; Takagi, I., On the shape of the least-energy solutions to a semilinear Neumann problem, Comm. pure appl. math., Vol. 44, 819-851, (1991) · Zbl 0754.35042
[11] Rey, O., The role of the Green’s function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. funct. anal., Vol. 89, 1-52, (1990) · Zbl 0786.35059
[12] \scJ. Wei, Viscosity approach to the construction of single-peaked solutions of a singularly perturbed semilinear Dirichlet problem, preprint. · Zbl 0865.35011
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.