# 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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1224-1784

1844-0835

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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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