Nonselfadjoint resonance problems with unbounded perturbations. (English) Zbl 0599.35069

Let L be an elliptic but not necessarily self-adjoint operator on \(\Omega \subset R^ n\) and g be locally Lipschitzian. This paper studies the solvability of the Dirichlet problem \[ Lu=\lambda_ 1u+g(u)-h;\quad u=0\quad on\quad \partial \Omega, \] under the growth assumption \(\limsup_{| \xi | \to \infty} g(\xi)/\xi <d_ 0-\lambda_ 1\) where \(d_ 0>\lambda_ 1\) reduces to \(\lambda_ 2\) when L is self- adjoint and \(\lambda_ 1\) is the principal eigenvalue of L, and the Landesman-Lazer condition \[ \limsup_{u\to -\infty} g(u)\int_{\Omega}\theta^*_ 1(x) dx<\int_{\Omega}\theta^*_ 1(x) h(x) dx< \limsup_{u\to +\infty} g(u)\int_{\Omega}\theta^*_ 1(x) dx, \] where \(\theta^*_ 1\) is strictly positive eigenfunction of \(L^*\) corresponding to \(\lambda_ 1\). The proof uses a Leray-Schauder type argument whose applicability depends upon an interesting auxiliary result on the solutions of an associated linear problem.
Reviewer: J.Mawhin


35J65 Nonlinear boundary value problems for linear elliptic equations
35B20 Perturbations in context of PDEs
35B50 Maximum principles in context of PDEs
Full Text: DOI


[1] Ahmad, S., A resonance problem in which the nonlinearity may grow linearly, Proc. Am. math. Soc., 92, 381-384 (1984) · Zbl 0562.34011
[2] Amann, H.; Crandall, M. G., On some existence theorems for semi-linear elliptic equations, Indiana Univ. Math. J., 27, 779-790 (1978) · Zbl 0391.35030
[3] Ambrosetti, A.; Mancini, G., Existence and multiplicity results for nonlinear elliptic problems with linear part at resonance—the case of the simple eigenvalue, J. diff. Eqns, 28, 220-245 (1978) · Zbl 0393.35032
[4] Bony, J. M., Principe de maximum dans les espaces de Sobolev, C.r. hebd. séanc. Acad. Sci. Paris, 265, Ser. 4, 333-336 (1967) · Zbl 0164.16803
[5] Brezis, H.; Nirenberg, L., Characterization of the ranges of some nonlinear operators and application to boundary value problems, Annali Scu. norm. sup. Pisa, 5, 225-326 (1978) · Zbl 0386.47035
[6] Cesari, L.; Kannan, R., Existence of solutions of a nonlinear differential equation, Proc. Am. math. Soc., 88, 605-613 (1983) · Zbl 0529.34005
[7] Fučik, S., Surjectivity of operators involving linear noninvertible part and nonlinear compact perturbation, Funkcialaj Ekvacioj, 17, 73-83 (1974) · Zbl 0294.47041
[8] Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (1977), Springer: Springer Berlin · Zbl 0691.35001
[9] Landesman, E. M.; Lazer, A. C., Nonlinear perturbations of a linear elliptic boundary value problem at resonance, J. Math. Mech., 19, 609-623 (1970) · Zbl 0193.39203
[10] Manes, A.; Micheletti, A. M., Un estensione della teoria variazionale classica degli autovalori per operatori ellittici del secondo ordine, Boll. Un. Mat. Ital., 7, 285-301 (1973) · Zbl 0275.49042
[11] MAwhinArdIllem; MAwhinArdIllem
[12] Protter, M. H.; Weinberger, H. F., Maximum Principles in Differential Equations (1967), Prentice Hall: Prentice Hall Englewood Cliffs, N.J · Zbl 0153.13602
[13] Sattinger, D. H., Topics in stability and bifurcation theory, (Lecture Notes in Mathematics, 309 (1973), Springer: Springer Berlin) · Zbl 0466.58016
[14] Schechter, M.; Shapiro, J.; Snow, M., Solutions of the nonlinear problem \(Au}=N(u)\) in a Banach space, Trans. Am. math. Soc., 241, 69-78 (1978) · Zbl 0403.47030
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