Advances in Differential Equations

An improved Poincaré inequality and the $p$-Laplacian at resonance for $p>2$

Jacqueline Fleckinger-Pellé and Peter Takáč

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


An improved Poincaré inequality is shown for $p>2$: There exists a constant $c>0$ such that for all $u\in W_0^{1,p}(\Omega)$, \begin{equation*} \tag*{(P)} \int_\Omega |\nabla u|^p \,{\rm d}x - \lambda_1 \int_\Omega |u|^p \,{\rm d}x \geq c \Big( | u^\parallel |^{p-2} \int_\Omega |\nabla\varphi_1|^{p-2} |\nabla u^\top|^2 \,{\rm d}x + \int_\Omega |\nabla u^\top|^p \,{\rm d}x \Big) . \end{equation*} Here, a function $u\in L^2(\Omega)$ is decomposed as an orthogonal sum \[ u = u^\parallel\cdot \varphi_1 + u^\top \;\mbox{ where }\; u^\parallel {\stackrel{{\mathrm {def}}}{=}} \|\varphi_1\|_{ L^2(\Omega) }^{-2} \langle u, \varphi_1 \rangle \;\mbox{ and }\; \langle u^\top, \varphi_1 \rangle = 0 , \] $\lambda_1$ denotes the first eigenvalue of the positive Dirichlet $p$-Laplacian $-\Delta_p$, $\Delta_p u\equiv {\mathop{\mathrm {div}}} ( |\nabla u|^{p-2} \nabla u )$, $\lambda_1$ is simple, and $\varphi_1$ stands for the corresponding eigenfunction. Inequality (P) is applied to show the existence of a weak solution to the following degenerate quasi\-linear boundary value problem at resonance, where $f\in L^2(\Omega)$ with $\langle f,\varphi_1 \rangle = 0$: \[ - \Delta_p u = \lambda_1 |u|^{p-2} u + f(x) \;\mbox{ in } \Omega ;\qquad u = 0 \;\mbox{ on } \partial\Omega . \]

Article information

Adv. Differential Equations, Volume 7, Number 8 (2002), 951-971.

First available in Project Euclid: 27 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J60: Nonlinear elliptic equations
Secondary: 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems


Fleckinger-Pellé, Jacqueline; Takáč, Peter. An improved Poincaré inequality and the $p$-Laplacian at resonance for $p>2$. Adv. Differential Equations 7 (2002), no. 8, 951--971.

Export citation