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Multiple existence of solutions for a nonlinear elliptic problem on a Riemannian manifold. (English) Zbl 1157.58006

Let \((M,g)\) be a compact, connected and orientable smooth \(N\)-dimensional Riemannian manifold without boundary, and let \(p\in(2,2^*)\) with \(2^*=2N/(N-2).\)
The author derives the rexistence of multiple solutions to the problem
\[ -\varepsilon^2\Delta_gu+u=| u| ^{p-2}u\quad \text{on}\;M \] with \(\Delta_g\) standing for the Laplacian operator over \(M.\)

MSC:

58J05 Elliptic equations on manifolds, general theory
35J65 Nonlinear boundary value problems for linear elliptic equations
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[1] Bahri, A.; Coron, M., On a nonlinear elliptic equation involving the critical sobolev exponent. the effect of the topology of the domain, Commun. Pure Appl. Math., 41, 253-294 (1988) · Zbl 0649.35033
[2] Bartsch, T.; Weth, Tobias, Three nodal solutions of singularly perturbed elliptic equations on domains without topology, Ann. Inst. H. Poincare, 22, 259-281 (2005) · Zbl 1114.35068
[3] V. Benci, C Bonanno, A.M. Micheletti, On the multiplicity of solutions of a nonlinear elliptic problem on Riemannian manifolds (in press); V. Benci, C Bonanno, A.M. Micheletti, On the multiplicity of solutions of a nonlinear elliptic problem on Riemannian manifolds (in press) · Zbl 1130.58010
[4] Benci, V.; Cerami, G., The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems, Arch. Ration. Mech. Anal., 119, 79-93 (1991) · Zbl 0727.35055
[5] Chavel, I., Rienmannian Geometry — A Modern Introduction (1993), Cambridge University Press
[6] Dancer, E. N.; Wei, J., On the effect of domain topology in a singular perturbation problem, Topol. Methods Nonlinear Anal., 11, 227-248 (1998) · Zbl 0926.35015
[7] Fournier, G.; Willem, M., Multiple solutions of the forced doule pendulum equation, Ann. Inst. H. Poincare Anal. non Lineaire, 6, 259-281 (1989) · Zbl 0683.70022
[8] Gidas, B.; Ni, W. M.; Nirenberg, L., Symmetry of positive solutions for nonlinear elliptic equation in \(r^n\), Adv. Math. Suppl. Stud., 7, 369-402 (1981)
[9] Hirano, N., Multiple existence of solutions for semilinear elliptic problems on a do main with a rich topology, Nonlinear Anal. TMA, 29, 725-736 (1997) · Zbl 0881.35043
[10] Hirano, N., A nonlinear elliptic equation with critical exponents: Effect of geometry and topology of the domain, J. Differential Equations, 182, 78-107 (2002) · Zbl 1014.35031
[11] Kwong, M. K., Uniqueness of positive solutions of \(- \delta u - u + u^p = 0\) in \(r^n\), Arch. Ration. Mech. Anal., 105, 243-266 (1989) · Zbl 0676.35032
[12] Passaseo, D., Multiplicity of positive solutions of nonlinear elliptic equations with critical sobolev exponent in some contractible domains, Manuscripta Math., 65, 147-175 (1989) · Zbl 0701.35068
[13] Cerami, G.; Passaseo, D., The effect of concentrating potentials in some singularly perturbed problems, Carc. Var. Partial Differential Equations, 17, 257-281 (2003) · Zbl 1290.35050
[14] Peterson, P., Riemannian Geometry, GTM, vol. 171 (1997), Springer
[15] Rosenberg, S., (The Laplacian on a Riemannian Manifold. The Laplacian on a Riemannian Manifold, Student Texts, vol. 31 (1997), London Mathematical Society) · Zbl 0868.58074
[16] Schwartz, J. T., Nonlinear Functional Analysis (1969), Gordon and Breach Science Publisher: Gordon and Breach Science Publisher New York · Zbl 0203.14501
[17] Szulkin, A., A relative category and applications to critical point theory for strongly indefinite functionals, Nonlinear Anal., 15, 725-739 (1990) · Zbl 0719.58011
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