A remark on the preceding paper of Fucik and Krbec. (English) Zbl 0356.35030

Hess, Peter

Math. Z. 155, 139-141 (1977).

The solvability of a nonlinear elliptic boundary value problem of the form \[ A u+g(u)=f \text { in } 8, \quad u=0 \text { on } \partial \Omega \text {, } \] is investigated under the assumptions that \( A \) is a linear, uniformly elliptic selfadjoint differential operator (of second order) with nontrivial nullspace \( N(A), f \in I^{\infty}(\Omega) \) is given, and \( g: \mathbb{R} \rightarrow \mathbb{R} \) is a continuous function with \( 0=g_{ \pm}:=\lim _{t \rightarrow \pm \infty} g(t) \). If \( g_{-} \neq g_{+} \), a necessary (and ”almost sufficient”) condition, due to Landesman and Lazer, is known. This condition is meaningless in the present case. In this note it is proved in a simple manner that a solution exists for \( f \perp N(A) \) (in \( L^{2}(8) \) ) provided liminf \( t \cdot g(t)>0 \). This is a slight \( t \rightarrow \pm \infty \) generalization of a result of S. \( \mathrm{F} u \check{c} i \mathrm{k} \) and \( \mathrm{M}. \mathrm{K} \mathrm{r} \mathrm{b} \mathrm{e} \mathrm{c} \) [Math. Z. 155, 129-138 (1977; Zbl. 337.35034)]. - (Remark. More general results in this direction are contained in the author’s paper ”Perturbations non lineaires de pro- 35031 blèmes linéaires à la résonance: existence de multiples solutions” [to appear in Proc. Journées d’Analyse non linéaire, Besançon 1977 (Springer Lecture Notes in Mathematics)].)

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MSC:

35J60

Nonlinear elliptic equations

References:

[1]	Friedman, A.: Partial Differential Equations. New York: Holt, Rinehart and Winston 1969 · Zbl 0224.35002
[2]	Fu?ik, S., Krbec, M.: Boundary value problems with bounded nonlinearity and general nullspace of the linear part. Math. Z.155, 129-138 (1977) · doi:10.1007/BF01214212
[3]	Mizohata, S.: The Theory of Partial Differential Equations. Cambridge: University Press 1973 · Zbl 0263.35001
[4]	Rabinowitz, P.: Some minimax theorems and applications to nonlinear partial differential equations. In: Nonlinear Analysis (dedicated to Prof. E. Roth on the occasion of his 80th birthday). New York-London: Academic Press (to appear)