zbMATH — the first resource for mathematics

Bifurcation from infinity and multiple solutions for some Dirichlet problems with unbounded nonlinearities. (English) Zbl 0780.35038
The authors study an elliptic problem of the form \[ -\Delta u-\lambda u= f(x,u)- h(x) \text{ in } \Omega, \qquad u=0 \text{ on } \partial\Omega. \tag{P} \] The nonlinearity \(f\) is sufficiently smooth and has sublinear growth at \(+\infty\), i.e. \(\lim_{s\to+\infty} {{f(x,s)} \over s} =0\) uniformly for \(x\in\Omega\). Moreover, \(\lambda\in\mathbb{R}\) and \(h\in C^{0,\alpha} (\overline {\Omega})\). The authors prove the following multiplicity result:
Theorem. Let \(\lambda_ 1\) denote the first eigenvalue of \(-\Delta\) and let \(\varphi\) denote a positive eigenfunction of \(-\Delta\) associated with \(\lambda_ 1\). Assume there exist \(c,C\in L^ 1(\Omega)\) such that \(c(x)\leq \liminf_{s\to -\infty} f(x,s)\), \(C(x)\geq \limsup_{s\to +\infty} f(x,s)\), uniformly for \(x\in\Omega\) and such that \[ \int_ \Omega c(x)\varphi(x)dx> \int_ \Omega h(x)\varphi(x)dx> \int_ \Omega C(x)\varphi(x)dx. \] Then there exists \(\delta>0\) such that (P) has at least one solution for \(\lambda\leq \lambda_ 1\) and at least two solutions for \(\lambda_ 1<\lambda<\lambda_ 1+\delta\).
The paper constitutes a development of J. Mawhin and K. Schmitt [Result. Math. 14, No. 1/2, 138-146 (1988; Zbl 0780.35043)]. It should be noted that no upper bound is imposed on \(f\), as \(s\to-\infty\). The technique is based on the use of super- and sub-solutions and the bifurcation theorem from infinity of P. H. Rabinowitz [J. Differ. Equations 14, 462-475 (1973; Zbl 0272.35017)].

35J65 Nonlinear boundary value problems for linear elliptic equations
35B32 Bifurcations in context of PDEs
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
Full Text: DOI
[1] Landesman, E.H.; Lazer, A.C., Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. math. mech., 19, 609-623, (1970) · Zbl 0193.39203
[2] Hess, P., On a theorem by landesman and lazer, Indiana univ. math. J., 23, 827-829, (1974) · Zbl 0259.35036
[3] Amann, H.; Ambrosetti, A.; Mancini, C., Elliptic equations with noninvertible Fredholm linear part and bounded nonlinearities, Math. Z., 158, 179-194, (1978) · Zbl 0368.35032
[4] Brezis, H.; Nirenberg, L., Characterizations of the ranges of some nonlinear operators and applications to boundary value problems, Annali scu. norm. sup. Pisa, 5, 14, 115-175, (1978)
[5] de Figueiredo, D.G., Semilinear elliptic equations at resonance: higher eigenvalues and unbounded nonlinearities, (), 89-99
[6] Mawhin, J., Landesman-Lazer’s type problem for nonlinear equations, Conf. semin. mat. univ. Bari, 147, 1-22, (1977) · Zbl 0436.47050
[7] Mawhin, J.; Schmitt, K., Landesman-lazer type problems at an eigenvalue of odd multiplicity, Results math., 14, 138-146, (1988) · Zbl 0780.35043
[8] Mawhin, J., Bifurcation from infinity and nonlinear boundary value problems, (), 119-129
[9] Mawhin, J.; Schmitt, K., Nonlinear eigenvalue problems with the parameter near resonance, Annls Pol. math., 51, 241-248, (1990) · Zbl 0724.34025
[10] Rabinowitz, P., On bifurcation from infinity, J. diff. eqns, 14, 462-475, (1973) · Zbl 0272.35017
[11] Kazdan, J.L.; Warner, F.W., Remarks on some quasilinear elliptic equations, Communs pure appl. math., 28, 567-597, (1975) · Zbl 0325.35038
[12] Badiale, M.; Lupo, D., Some remarks on a multiplicity result by mawhin and schmitt, Bull. acad. r. belg. cl. sci., 65, 5, 210-224, (1989) · Zbl 0706.34020
[13] Adams, R.A., Sobolev spaces, (1975), Academic Press New York · Zbl 0186.19101
[14] Stampacchia, G., Equations elliptiques du second ordre à coefficients discontinus, Séminaire de mathématiques supérieures, Vol. 16, (1966) · Zbl 0151.15501
[15] de Figueiredo, D.G., Positive solutions of semilinear elliptic problems, (), 34-87
[16] Mawhin, J., Points fixes, points critiques et problèmes aux limites, Séminaire de mathématique supérieures, Vol. 92, (1985) · Zbl 0561.34001
[17] Agmon, S., The L_{p} approach to the Dirichlet problem, Annali scu. norm. sup. Pisa, 13, 405-448, (1959) · Zbl 0093.10601
[18] Bondarenko, V.A.; Zabreiko, P.P., The superposition operator in Hölder function spaces, Soviet math. dokl., 16, 739-743, (1975) · Zbl 0328.47040
[19] Protter, M.H.; Weinberger, H.F., Maximum principles in differential equations, (1967), Prentice-Hall Englewood Cliffs, New Jersey · Zbl 0153.13602
[20] Rabinowitz, P., Some global results for nonlinear eigenvalue problems, J. funct. analysis, 7, 487-513, (1971) · Zbl 0212.16504
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.