Differential and Integral Equations

On the closed solution to some nonhomogeneous eigenvalue problems with $p$-Laplacian

Pavel Drábek and Raúl Manásevich

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We deal with the Dirichlet, Neumann and periodic eigenvalue problems for the equation $$ \quad(|u'|^{p-2}u')' +\lambda |u|^{q-2}u=0, \text{\quad on~~}(0,T), $$ where $T>0,$ $\lambda>0,$ and $p,q>1.$ For those problems we obtain a complete description of the spectra and a closed form representation of the corresponding eigenfunctions. As an application of our results we present sharp Poincar\'e and Wirtinger inequalities for the imbeddings $W_0^{1,p}(0,T)$ into $L^q(0,T)$ and $W_T^{1,p}(0,T)$ into $L^q(0,T),$ respectively.

Article information

Differential Integral Equations, Volume 12, Number 6 (1999), 773-788.

First available in Project Euclid: 29 April 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34B15: Nonlinear boundary value problems
Secondary: 34C25: Periodic solutions 34L30: Nonlinear ordinary differential operators 47J30: Variational methods [See also 58Exx]


Drábek, Pavel; Manásevich, Raúl. On the closed solution to some nonhomogeneous eigenvalue problems with $p$-Laplacian. Differential Integral Equations 12 (1999), no. 6, 773--788. https://projecteuclid.org/euclid.die/1367241475

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