# The multiplicity of solutions and geometry of a nonlinear elliptic equation

Q. Choi; Sungki Chun; Tacksun Jung

Studia Mathematica (1996)

- Volume: 120, Issue: 3, page 259-270
- ISSN: 0039-3223

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topChoi, Q., Chun, Sungki, and Jung, Tacksun. "The multiplicity of solutions and geometry of a nonlinear elliptic equation." Studia Mathematica 120.3 (1996): 259-270. <http://eudml.org/doc/216336>.

@article{Choi1996,

abstract = {Let Ω be a bounded domain in $ℝ^n$ with smooth boundary ∂Ω and let L denote a second order linear elliptic differential operator and a mapping from $L^2(Ω)$ into itself with compact inverse, with eigenvalues $-λ_\{i\}$, each repeated according to its multiplicity, 0 < λ1 < λ2 < λ3 ≤ ... ≤ λi ≤ ... → ∞. We consider a semilinear elliptic Dirichlet problem $Lu+bu^+-au^-=f(x)$ in Ω, u=0 on ∂ Ω. We assume that $a < λ_\{1\}$, $λ_\{2\} < b < λ_\{3\}$ and f is generated by $ϕ_\{1\}$ and $ϕ_\{2\}$. We show a relation between the multiplicity of solutions and source terms in the equation.},

author = {Choi, Q., Chun, Sungki, Jung, Tacksun},

journal = {Studia Mathematica},

keywords = {multiplicity; semilinear Dirichlet problem; cone},

language = {eng},

number = {3},

pages = {259-270},

title = {The multiplicity of solutions and geometry of a nonlinear elliptic equation},

url = {http://eudml.org/doc/216336},

volume = {120},

year = {1996},

}

TY - JOUR

AU - Choi, Q.

AU - Chun, Sungki

AU - Jung, Tacksun

TI - The multiplicity of solutions and geometry of a nonlinear elliptic equation

JO - Studia Mathematica

PY - 1996

VL - 120

IS - 3

SP - 259

EP - 270

AB - Let Ω be a bounded domain in $ℝ^n$ with smooth boundary ∂Ω and let L denote a second order linear elliptic differential operator and a mapping from $L^2(Ω)$ into itself with compact inverse, with eigenvalues $-λ_{i}$, each repeated according to its multiplicity, 0 < λ1 < λ2 < λ3 ≤ ... ≤ λi ≤ ... → ∞. We consider a semilinear elliptic Dirichlet problem $Lu+bu^+-au^-=f(x)$ in Ω, u=0 on ∂ Ω. We assume that $a < λ_{1}$, $λ_{2} < b < λ_{3}$ and f is generated by $ϕ_{1}$ and $ϕ_{2}$. We show a relation between the multiplicity of solutions and source terms in the equation.

LA - eng

KW - multiplicity; semilinear Dirichlet problem; cone

UR - http://eudml.org/doc/216336

ER -

## References

top- [1] H. Amann and P. Hess, A multiplicity result for a class of elliptic boundary value problems, Proc. Roy. Soc. Edinburgh 84 (1979), 145-151. Zbl0416.35029
- [2] A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis, Cambridge Stud. Adv. Math. 34, Cambridge University Press, 1993. Zbl0781.47046
- [3] Q. H. Choi and T. Jung, Multiplicity of solutions of nonlinear wave equations with nonlinearities crossing eigenvalues, Hokkaido Math. J. 24 (1995), 53-62. Zbl0835.35091
- [4] Q. H. Choi and T. Jung, An application of a variational reduction method to a nonlinear wave equation, J. Differential Equations 117 (1995), 390-410. Zbl0830.35070
- [5] J. M. Coron, Periodic solutions of a nonlinear wave equation without assumptions of monotonicity, Math. Ann. 262 (1983), 273-285.
- [6] E. N. Dancer, On the ranges of certain weakly nonlinear elliptic partial differential equations, J. Math. Pures Appl. 57 (1978), 351-366. Zbl0394.35040
- [7] A. C. Lazer and P. J. McKenna, Existence, uniqueness, and stability of oscillations in differential equations with symmetric nonlinearities, Trans. Amer. Math. Soc. 315 (1989), 721-739. Zbl0725.34042
- [8] A. C. Lazer and P. J. McKenna, Some multiplicity results for a class of semilinear elliptic and parabolic boundary value problems, J. Math. Anal. Appl. 107 (1985), 371-395. Zbl0584.35053
- [9] A. C. Lazer and P. J. McKenna, A symmetry theorem and applications to nonlinear partial differential equations, J. Differential Equations 72 (1988), 95-106. Zbl0666.47038
- [10] P. J. McKenna, Topological Methods for Asymmetric Boundary Value Problems, Lecture Notes Ser. 11, Res. Inst. Math., Global Analysis Res. Center, Seoul National University, 1993. Zbl0944.35001
- [11] P. J. McKenna, R. Redlinger and W. Walter, Multiplicity results for asymptotically homogeneous semilinear boundary value problems, Ann. Mat. Pura Appl. (4) 143 (1986), 347-257. Zbl0607.35038
- [12] P. J. McKenna and W. Walter, On the multiplicity of the solution set of some nonlinear boundary value problems, Nonlinear Anal. 8 (1984), 893-907. Zbl0556.35024
- [13] K. Schmitt, Boundary value problems with jumping nonlinearities, Rocky Mountain J. Math. 16 (1986), 481-496. Zbl0631.35032
- [14] J. Schröder, Operator Inequalities, Academic Press, New York, 1980.
- [15] S. Solimini, Some remarks on the number of solutions of some nonlinear elliptic problems, Ann. Inst. H. Poincaré 2 (1985), 143-156. Zbl0583.35044

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