×

zbMATH — the first resource for mathematics

Positive solutions to a class of elliptic boundary value problems. (English) Zbl 0915.35043
Summary: We prove the existence of positive solutions to the boundary value problems \[ \Delta u+\lambda a(x)f(u)= 0\quad\text{in }\Omega,\quad u= 0\quad\text{on }\partial\Omega, \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(a:\Omega\to \mathbb{R}\) may change sign, \(f(0)>0\), and \(\lambda> 0\) is sufficiently small. Our approach is based on the Leray-Schauder fixed point theorem. \(\copyright\) Academic Press.

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Cac, N.P.; Fink, A.M.; Gatica, J.A., Nonnegative solutions of the radial Laplacian with nonlinearity that changes sign, Proc. amer. math. soc., 123, 1393-1398, (1995) · Zbl 0826.34021
[2] Courant, R.; Hilbert, D., Methods of mathematical physics, (1953), Interscience New York · Zbl 0729.00007
[3] Hess, P., On bifurcation and stability of positive solutions of nonlinear elliptic eigenvalue problems, (), 103-119
[4] Hess, P.; Kato, T., On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. partial differential equations, 5, 999-1030, (1980) · Zbl 0477.35075
[5] Lions, P.L., On the existence of positive solutions of semilinear elliptic equations, SIAM rev., 24, 441-467, (1982) · Zbl 0511.35033
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.