Variational methods for a resonant problem with the $p$-Laplacian in $\Bbb R^N$

Alziary, Benedicte ; Fleckinger, Jacqueline ; Takáč, Peter

Electronic Journal of Differential Equations, Tome 2004 (2004).

Résumé

Summary: 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 mathbbR^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, $p$ 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 $C_{\rm c}^1(\mathbb{R}^N)$ 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 $1 less than p less than 2$, and an improved Poincare inequality if $2\leq p less than N$. 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.

Détail

Classification : 35P30, 35J20, 47J10, 47J30
Keywords: p-Laplacian, degenerate quasilinear Cauchy problem, Fredholm alternative, (p-1)-homogeneous problem at resonance, saddle point geometry, improved Poincarè inequality, second-order Taylor formula