Electronic Journal of Differential Equations,
Vol. 2004(2004), No. 76, pp. 1-32.
Title: Variational methods for a resonant problem
with the p-Laplacian in $\mathbb{R}^N$
Authors: Benedicte Alziary (Univ. Toulouse 1, France)
Jacqueline Fleckinger (Univ. Toulouse 1, France)
Peter Takac (Univ. Rostock, Germany)
Abstract:
The solvability of the resonant Cauchy problem
$$
- \Delta_p u = \lambda_1 m(|x|) |u|^{p-2} u + f(x)
\quad\hbox{in } \mathbb{R}^N ;\quad u\in D^{1,p}(\mathbb{R}^N),
$$
in the entire Euclidean space $\mathbb{R}^N$ ($N\geq 1$) is
investigated as a part of the Fredholm alternative at the first
(smallest) eigenvalue $\lambda_1$ of the positive $p$-Laplacian
$-\Delta_p$ on $D^{1,p}(\mathbb{R}^N)$ relative to the weight
$m(|x|)$. Here, $\Delta_p$ stands for the $p$-Laplacian,
$m\colon \mathbb{R}_+\to \mathbb{R}_+$ is a weight
function assumed to be radially symmetric,
$m\not\equiv 0$ in $\mathbb{R}_+$, and
$f\colon \mathbb{R}^N\to \mathbb{R}$ is a
given function satisfying a suitable integrability condition.
The weight $m(r)$ is assumed to be bounded and to decay fast enough
as $r\to +\infty$. Let $\varphi_1$ denote the (positive)
eigenfunction associated with the (simple) eigenvalue $\lambda_1$
of $-\Delta_p$. If $\int_{\mathbb{R}^N} f\varphi_1 \,{\rm d}x =0$,
we show that problem has at least one solution $u$
in the completion $D^{1,p}(\mathbb{R}^N)$ of
$C_{\mathrm{c}}^1(\mathbb{R}^N)$ endowed with the norm
$(\int_{\mathbb{R}^N} |\nabla u|^p \,{\rm d}x)^{1/p}$.
To establish this existence result, we employ a saddle point method
if $1 < p < 2$, and an improved Poincar\'e inequality if
$2\leq p< N$. We use weighted Lebesgue and Sobolev spaces with
weights depending on $\varphi_1$. The asymptotic behavior of
$\varphi_1(x)= \varphi_1(|x|)$ as $|x|\to \infty$ plays a crucial role.
\end{abstract}
Submitted March 19, 2004. Published May 26, 2004.
Math Subject Classifications: 35P30, 35J20, 47J10, 47J30
Key Words: p-Laplacian; degenerate quasilinear Cauchy problem;
Fredholm alternative; (p-1)-homogeneous problem at resonance;
saddle point geometry; improved Poincare inequality;
second-order Taylor formula.