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Search: MSC category 35J70 ( Degenerate elliptic equations )

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1. CMB Online first

Cruz-Uribe, David; Rodney, Scott; Rosta, Emily
 PoincarÃ© Inequalities and Neumann Problems for the $p$-Laplacian We prove an equivalence between weighted PoincarÃ© inequalities and the existence of weak solutions to a Neumann problem related to a degenerate $p$-Laplacian. The PoincarÃ© inequalities are formulated in the context of degenerate Sobolev spaces defined in terms of a quadratic form, and the associated matrix is the source of the degeneracy in the $p$-Laplacian. Keywords:degenerate Sobolev space, $p$-Laplacian, PoincarÃ© inequalitiesCategories:30C65, 35B65, 35J70, 42B35, 42B37, 46E35

2. CMB 2006 (vol 49 pp. 358)

Khalil, Abdelouahed El; Manouni, Said El; Ouanan, Mohammed
 On the Principal Eigencurve of the $p$-Laplacian: Stability Phenomena We show that each point of the principal eigencurve of the nonlinear problem $$-\Delta_{p}u-\lambda m(x)|u|^{p-2}u=\mu|u|^{p-2}u \quad \text{in } \Omega,$$ is stable (continuous) with respect to the exponent $p$ varying in $(1,\infty)$; we also prove some convergence results of the principal eigenfunctions corresponding. Keywords:$p$-Laplacian with indefinite weight, principal eigencurve, principal eigenvalue, principal eigenfunction, stabilityCategories:35P30, 35P60, 35J70

3. CMB 1997 (vol 40 pp. 244)

Naito, Yūki; Usami, Hiroyuki
 Nonexistence results of positive entire solutions for quasilinear elliptic inequalities This paper treats the quasilinear elliptic inequality $$\div (|Du|^{m-2}Du) \geq p(x)u^{\sigma}, \quad x \in \Rs^N,$$ where $N \geq 2$, $m > 1$, $\sigma > m - 1$, and $p \colon \Rs^N \rightarrow (0, \infty)$ is continuous. Sufficient conditions are given for this inequality to have no positive entire solutions. When $p$ has radial symmetry, the existence of positive entire solutions can be characterized by our results and some known results. Categories:35J70, 35B05
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