We study the mappings of monotone type in Oriicz-Sobolev spaces. We introduce a new class (Sm) as a generalization of (S+) and extend the definition of quasimonotone map. We also prove existence results for equations involving monotone-like mappings.

Our main result is a maximum principle bounding the absolute values of the solution in terms of the supremum of the absolute values of the boundary data.

Let m : [0, x [ --> [0, x [ be an increasing continuous function with m(t) = 0 if and only if t = 0. m(t) --> x as t --> x and Omega subset of R-N a bounded domain. In this paper we show that for every r > 0 the problem[GRAPHICS]has an infinite numher of eigenfunctions on the level set integral(Omega) M( \del u \ ) = r, where M(t) = integral(0)(\ t \) m(s) ds and g : R --> R is odd satisfying some growth condition. Moreover, we show that the sequence of associated eigenvalues tends to infinity. We emphasize that no Delta(2)-condition is needed for M or for its conjugate, so the associated functionals are not continuously differentiable, in general. (C) 2000 Academic Press.

We consider a nonlinear eigenvalue problem for some elliptic equations governed by general operators including the p-Laplacian. The natural framework in which we consider such equations is that of Orlicz–Sobolev spaces. We exhibit two positive constants λ0 and λ1 with λ0≤λ1 such that λ1 is an eigenvalue of the problem while any value λ<λ0 cannot be so. By means of Harnack-type inequalities and a strong maximum principle, we prove the isolation of λ1 on the right side. We emphasize that throughout the paper no Δ2-condition is needed.

We study the mappings of monotone type in Orlicz-Sobolev spaces. We introduce a new class $(S_m)$ as a generalization of $(S_+)$ and extend the definition of quasimonotone map. We also prove existence results for equations involving monotone-like mappings.