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Infinitely many radial solutions of a non--homogeneous $p$--Laplacian problem

Pages: 51 - 59, Issue special, November 2013

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Rossella Bartolo - Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, Via E. Orabona 4, 70125 Bari, Italy (email)
Anna Maria Candela - Dipartimento di Matematica, Università degli Studi di Bari "Aldo Moro", Via E. Orabona 4, 70125 Bari, Italy (email)
Addolorata Salvatore - Dipartimento di Matematica, Università degli Studi di Bari "Aldo Moro", Via E. Orabona 4, 70125 Bari, Italy (email)

Abstract: In this paper we investigate the existence of infinitely many radial solutions for the elliptic Dirichlet problem \[ \left\{ \begin{array}{ll} \displaystyle{-\Delta_p u\ =|u|^{q-2}u + f(x)} & \mbox{ in } B_R,\\ \displaystyle{u=\xi} & \mbox{ on } \partial B_R,\\ \end{array} \right. \] where $B_R$ is the open ball centered in $0$ with radius $R$ in $\mathbb{R}^N$ ($N \geq 3$), $2 < p < N$, $p< q < p^*$ (with $p^* = \frac{pN}{N-p}$), $\xi\in\mathbb{R}$ and $f$ is a continuous radial function in $\overline B_R$. The lack of even symmetry for the related functional is overcome by using some perturbative methods and the radial assumptions allow us to improve some previous results.

Keywords:  $p$--Laplacian operator, non--null boundary data, lack of symmetry, radial solution, variational approach, perturbative methods.
Mathematics Subject Classification:  Primary: 35J92; Secondary: 35J60, 35J66, 58E05.

Received: September 2012; Published: November 2013.

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