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.