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

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 ( 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 L u + b u + - a u - = 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.

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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  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
  12. [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. [13] K. Schmitt, Boundary value problems with jumping nonlinearities, Rocky Mountain J. Math. 16 (1986), 481-496. Zbl0631.35032
  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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