Electronic Journal of Differential Equations,
Vol. 2003(2003), No. 124, pp. 1-22.

Title: Resonance and strong resonance for semilinear elliptic equations
       in R^N

Authors: Gabriel Lopez Garza (Claremont Graduate Univ., California, USA)
         Adolfo J. Rumbos (Pomona College, California, USA)

Abstract: 
 We prove the existence of weak solutions for the semilinear 
 elliptic problem
 $$
 -\Delta u=\lambda hu+ag(u)+f,\quad u\in \mathcal{D}^{1,2}({\mathbb{R}^N}),
 $$
 where $\lambda \in \mathbb{R}$, $f\in L^{2N/(N+2)}$,
 $g:\mathbb{R} \to \mathbb{R}$ 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 $f\equiv 0$
 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.

Submitted June 3, 2003. Published December 16, 2003.
Math Subject Classifications: 35J20.
Key Words: Resonance; strong resonance; concentration-compactness.