On the bifurcation of the null solutions of some boundary value problems

Abstract

Existence and aymptotic expansion of some bifurcated solution for the following boundary value problem:

\begin{align*} -\Delta u+cu^{2}-B^{2}u & =0,\ \ \ \ \ \ \ x\in \Omega \\ u & =0,\ \ \ \ \ \ \ \partial x\in \Omega \end{align*}

are provided via the Lyapunov-Schmidt method and contraction mapping theorem. The bifurcation point occurs at the first eigenvalue of \(-\Delta\) operator.

Authors

C.I. Gheorghiu
Tiberiu Popoviciu Institute of Numerical Analysis

Keywords

nonlinear elliptic BVP; bifurcation point; Lyapunov-Schmidt method; contractions; asymptotic expansion

References

See the expanding block below.

Paper coordinates

C.I. Gheorghiu, Al. Tămăşan, On the bifurcation of the null solution of some mildly nonlinear elliptic boundary value problems, An. St. Univ. Ovidius Constanta, Seria Matematica, 5 (1996), 59-64.

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Paper on journal website
Print ISSN

1224-1784

Online ISSN

1844-0835

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References

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References

[1] C.I. Gheorghiu, Al., Tamasan, On the existence and uniqueness of some mildly nonlinear eliptic boundary value problems,  Rev. Num. Th. Approx., Cluj-Napoca, 1995.

[2] L.F. Shampine, G.M. Wing,  Existence and uniqueness of solutions of a class of nonlinear eliptic boundary value problems,  J. Math.Mec. 19, 1970, 971-979

[3] I. Stakgold,  branching of solution of nonlinear equations, SIAM, Review, 13, no.3, 1971, 289-332.

[4] P.L. Lions,  On the existence of positive solutions of semilinear eliptic equations, SIAM Review, 24, no.4, 1982, 441-467.

[5] H.M.Protter, H.F., Weinberger, Maximum principles in differential equations, Prentice Hasll Inc. 1967.

[6] M.S.Berger, L.F. Fraenkel,  On the asymptotic soluiton of a nonlinear Dirichlet problem, J. Math. Mech., 19, 1970, 553-585.

[7] L.V. Kantorovich, G.P. Akilov,  Analiza Functionala, Ed. Stiintifica si Enciclopedica, 1986.

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