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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)].

MSC:
 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
Citations:
Zbl 0780.35043; Zbl 0272.35017
Full Text:
References:
  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  Hess, P., On a theorem by landesman and lazer, Indiana univ. math. J., 23, 827-829, (1974) · Zbl 0259.35036  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  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)  de Figueiredo, D.G., Semilinear elliptic equations at resonance: higher eigenvalues and unbounded nonlinearities, (), 89-99  Mawhin, J., Landesman-Lazer’s type problem for nonlinear equations, Conf. semin. mat. univ. Bari, 147, 1-22, (1977) · Zbl 0436.47050  Mawhin, J.; Schmitt, K., Landesman-lazer type problems at an eigenvalue of odd multiplicity, Results math., 14, 138-146, (1988) · Zbl 0780.35043  Mawhin, J., Bifurcation from infinity and nonlinear boundary value problems, (), 119-129  Mawhin, J.; Schmitt, K., Nonlinear eigenvalue problems with the parameter near resonance, Annls Pol. math., 51, 241-248, (1990) · Zbl 0724.34025  Rabinowitz, P., On bifurcation from infinity, J. diff. eqns, 14, 462-475, (1973) · Zbl 0272.35017  Kazdan, J.L.; Warner, F.W., Remarks on some quasilinear elliptic equations, Communs pure appl. math., 28, 567-597, (1975) · Zbl 0325.35038  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  Adams, R.A., Sobolev spaces, (1975), Academic Press New York · Zbl 0186.19101  Stampacchia, G., Equations elliptiques du second ordre à coefficients discontinus, Séminaire de mathématiques supérieures, Vol. 16, (1966) · Zbl 0151.15501  de Figueiredo, D.G., Positive solutions of semilinear elliptic problems, (), 34-87  Mawhin, J., Points fixes, points critiques et problèmes aux limites, Séminaire de mathématique supérieures, Vol. 92, (1985) · Zbl 0561.34001  Agmon, S., The L_{p} approach to the Dirichlet problem, Annali scu. norm. sup. Pisa, 13, 405-448, (1959) · Zbl 0093.10601  Bondarenko, V.A.; Zabreiko, P.P., The superposition operator in Hölder function spaces, Soviet math. dokl., 16, 739-743, (1975) · Zbl 0328.47040  Protter, M.H.; Weinberger, H.F., Maximum principles in differential equations, (1967), Prentice-Hall Englewood Cliffs, New Jersey · Zbl 0153.13602  Rabinowitz, P., Some global results for nonlinear eigenvalue problems, J. funct. analysis, 7, 487-513, (1971) · Zbl 0212.16504
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