Electron. J. Diff. Eqns., Vol. 2004(2004), No. 76, pp. 1-32.

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

Benedicte Alziary, Jacqueline Fleckinger, Peter Takac

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, $p$ 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.

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.

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  Benedicte Alziary
CEREMATH & UMR MIP
Universite Toulouse 1 - Sciences Sociales
21 allees de Brienne, F-31042 Toulouse Cedex, France
email: alziary@univ-tlse1.fr
Jacqueline Fleckinger
CEREMATH & UMR MIP
Universite Toulouse 1 - Sciences Sociales
21 allees de Brienne, F-31042 Toulouse Cedex, France
e-mail: jfleck@univ-tlse1.fr
Peter Takac
Fachbereich Mathematik, Universitat Rostock
Universitatsplatz 1, D-18055 Rostock, Germany
email: peter.takac@mathematik.uni-rostock.de

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