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Search: All articles in the CMB digital archive with keyword $p$-Laplacian

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1. CMB Online first

Song, Hongxue; Chen, Caisheng; Yan, Qinglun
Existence of multiple solutions for a $p$-Laplacian system in $\textbf{R}^{N}$ with sign-changing weight functions
In this paper, we consider the quasi-linear elliptic problem \[ \left\{ \begin{aligned} & -M \left(\int_{\mathbb{R}^{N}}|x|^{-ap}|\nabla u|^{p}dx \right){\rm div} \left(|x|^{-ap}|\nabla u|^{p-2}\nabla u \right) \\ & \qquad=\frac{\alpha}{\alpha+\beta}H(x)|u|^{\alpha-2}u|v|^{\beta}+\lambda h_{1}(x)|u|^{q-2}u, \\ & -M \left(\int_{\mathbb{R}^{N}}|x|^{-ap}|\nabla v|^{p}dx \right){\rm div} \left(|x|^{-ap}|\nabla v|^{p-2}\nabla v \right) \\ & \qquad=\frac{\beta}{\alpha+\beta}H(x)|v|^{\beta-2}v|u|^{\alpha}+\mu h_{2}(x)|v|^{q-2}v, \\ &u(x)\gt 0,\quad v(x)\gt 0, \quad x\in \mathbb{R}^{N} \end{aligned} \right. \] where $\lambda, \mu\gt 0$, $1\lt p\lt N$, $1\lt q\lt p\lt p(\tau+1)\lt \alpha+\beta\lt p^{*}=\frac{Np}{N-p}$, $0\leq a\lt \frac{N-p}{p}$, $a\leq b\lt a+1$, $d=a+1-b\gt 0$, $M(s)=k+l s^{\tau}$, $k\gt 0$, $l, \tau\geq0$ and the weight $H(x), h_{1}(x), h_{2}(x)$ are continuous functions which change sign in $\mathbb{R}^{N}$. We will prove that the problem has at least two positive solutions by using the Nehari manifold and the fibering maps associated with the Euler functional for this problem.

Keywords:Nehari manifold, quasilinear elliptic system, $p$-Laplacian operator, concave and convex nonlinearities

2. CMB 2008 (vol 51 pp. 217)

Filippakis, Michael E.; Papageorgiou, Nikolaos S.
A Multivalued Nonlinear System with the Vector $p$-Laplacian on the Semi-Infinity Interval
We study a second order nonlinear system driven by the vector $p$-Laplacian, with a multivalued nonlinearity and defined on the positive time semi-axis $\mathbb{R}_+.$ Using degree theoretic techniques we solve an auxiliary mixed boundary value problem defined on the finite interval $[0,n]$ and then via a diagonalization method we produce a solution for the original infinite time-horizon system.

Keywords:semi-infinity interval, vector $p$-Laplacian, multivalued nonlinear, fixed point index, Hartman condition, completely continuous map

3. CMB 2007 (vol 50 pp. 356)

Filippakis, Michael E.; Papageorgiou, Nikolaos S.
Existence of Positive Solutions for Nonlinear Noncoercive Hemivariational Inequalities
In this paper we investigate the existence of positive solutions for nonlinear elliptic problems driven by the $p$-Laplacian with a nonsmooth potential (hemivariational inequality). Under asymptotic conditions that make the Euler functional indefinite and incorporate in our framework the asymptotically linear problems, using a variational approach based on nonsmooth critical point theory, we obtain positive smooth solutions. Our analysis also leads naturally to multiplicity results.

Keywords:$p$-Laplacian, locally Lipschitz potential, nonsmooth critical point theory, principal eigenvalue, positive solutions, nonsmooth Mountain Pass Theorem
Categories:35J20, 35J60, 35J85

4. 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, stability
Categories:35P30, 35P60, 35J70

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