| Summary: | We prove the existence of weak solutions for the semilinear elliptic problem $$ -Delta u=lambda hu+ag(u)+f,quad uin mathcal{D}^{1,2}({mathbb{R}^N}), $$ where $lambda in mathbb{R}$, $fin L^{2N/(N+2)}$, $g:mathbb{R} o mathbb{R}$ is a continuous bounded function, and $h in L^{N/2}cap L^{alpha}$, $alpha>N/2$. We assume that $a in L^{2N/(N+2)}cap L^{infty}$ in the case of resonance and that $a in L^1 cap L^{infty}$ and $fequiv 0$ for the case of strong resonance. We prove first that the Palais-Smale condition holds for the functional associated with the semilinear problem using the concentration-compactness lemma of Lions. Then we prove the existence of weak solutions by applying the saddle point theorem of Rabinowitz for the cases of non-resonance and resonance, and a linking theorem of Silva in the case of strong resonance. The main theorems in this paper constitute an extension to $mathbb{R}^N$ of previous results in bounded domains by Ahmad, Lazer, and Paul [2], for the case of resonance, and by Silva [15] in the strong resonance case.
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