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## Studia Mathematica

1996 | 120 | 3 | 259-270
Tytuł artykułu

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

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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 < λ_{1} < λ_{2} < λ_{3} ≤ ... ≤ λ_{i} ≤ ... → ∞. 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.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
259-270
Opis fizyczny
Daty
wydano
1996
otrzymano
1995-01-04
poprawiono
1996-03-28
Twórcy
autor
• Department of Mathematics, Inha University, Incheon 402-751, Korea
autor
• Department of Mathematics, Inha University, Incheon 402-751, Korea
autor
• Department of Mathematics, Kunsan National University, Kunsan 573-360, Korea
Bibliografia
• [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.
• [2] A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis, Cambridge Stud. Adv. Math. 34, Cambridge University Press, 1993.
• [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.
• [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.
• [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.
• [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.
• [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.
• [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.
• [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.
• [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.
• [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.
• [13] K. Schmitt, Boundary value problems with jumping nonlinearities, Rocky Mountain J. Math. 16 (1986), 481-496.
• [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.
Typ dokumentu
Bibliografia
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