## Advances in Differential Equations

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

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

Jacqueline Fleckinger-Pellé and Peter Takáč

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 27 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1356651685

**Mathematical Reviews number (MathSciNet)**

MR1895113

**Zentralblatt MATH identifier**

1208.35049

**Subjects**

Primary: 35J60: Nonlinear elliptic equations

Secondary: 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems

#### Citation

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. https://projecteuclid.org/euclid.ade/1356651685