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

Studia Mathematica (1996)

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

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## Abstract

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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 ${L}^{2}\left(\Omega \right)$ into itself with compact inverse, with eigenvalues $-{\lambda }_{i}$, each repeated according to its multiplicity, 0 < λ1 < λ2 < λ3 ≤ ... ≤ λi ≤ ... → ∞. We consider a semilinear elliptic Dirichlet problem $Lu+b{u}^{+}-a{u}^{-}=f\left(x\right)$ in Ω, u=0 on ∂ Ω. We assume that $a<{\lambda }_{1}$, ${\lambda }_{2} and f is generated by ${\varphi }_{1}$ and ${\varphi }_{2}$. We show a relation between the multiplicity of solutions and source terms in the equation.

## How to cite

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Choi, 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

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10. [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. [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
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14. [14] J. Schröder, Operator Inequalities, Academic Press, New York, 1980.
15. [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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