Resonance and strong resonance for semilinear elliptic equations in RN

In this dissertation we prove the existence of weak solutions for the semilinear elliptic equation −Δu = λ hu + ag(u), u ∈ $D1,2RN$, where λ ∈ $R$, g : $R$ → $R$ is a continuous, bounded function, h ∈ $LN2$ ∩ Lα, α > N/2 and a ∈ $L2NN+2$ ∩ L∞, for the case of resonance, a ∈ L1 ∩ L ∞ for the case of strong resonance. We prove first that the Palais-Smale condition holds for the functional associated with the semilinear problem, by the use of the Concentration-Compactness Lemma of Lions. Then we prove the existence of weak, solutions for the non-resonance and resonance cases by the use of the Saddle Point Theorem of Rabinowitz and by the use of a linking theorem in the case of strong resonance. The main theorems in this thesis are an extension to $RN$ of previous results in bounded domains by Ahmad, Lazer and Paul, for the resonance case, and Silva, for the strong resonance case.