Abstract
In this paper we deal with semilinear problems at resonance. We present a sufficient condition for the existence of a weak solution in terms of the asymptotic properties of nonlinearity. Our condition generalizes the classical Landesman-Lazer condition but it also covers the cases of vanishing and oscillating nonlinearities.
Highlights
Let Ω ⊆ Rn be a bounded domain, g : R → R be a bounded continuous function and f ∈ L2(Ω)
The purpose of this paper is to introduce a rather general sufficient condition of the Landesman–Lazer type for the existence of a solution of (1.1)
We proved that (LL)± imply (SC)±
Summary
Let Ω ⊆ Rn be a bounded domain, g : R → R be a bounded continuous function and f ∈ L2(Ω). We prove existence results even in this case. The above mentioned case g(s) = arctan s + c · cos s is covered by the so called potential Landesman–Lazer condition: G∓ φ+ dx − G± φ− dx < fφ dx < G± φ+ dx − G∓ φ− dx l’Hospital’s rule implies G− It follows from our Theorem 1.1 that (1.1) with g given above has a solution for any f ∈ L2(Ω)⊥.
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