Multiplicity of positive solutions for $(p,q)$Laplace equations with two parameters
Abstract
We study the zero Dirichlet problem for the equation $\Delta_p u \Delta_q u = \alpha u^{p2}u+\beta u^{q2}u$ in a bounded domain $\Omega \subset \mathbb{R}^N$, with $1<q<p$. We investigate the relation between two critical curves on the $(\alpha,\beta)$plane corresponding to the threshold of existence of special classes of positive solutions. In particular, in certain neighbourhoods of the point $(\alpha,\beta) = \left(\\nabla \varphi_p\_p^p/\\varphi_p\_p^p, \\nabla \varphi_p\_q^q/\\varphi_p\_q^q\right)$, where $\varphi_p$ is the first eigenfunction of the $p$Laplacian, we show the existence of two and, which is rather unexpected, three distinct positive solutions, depending on a relation between the exponents $p$ and $q$.
 Publication:

arXiv eprints
 Pub Date:
 July 2020
 arXiv:
 arXiv:2007.11623
 Bibcode:
 2020arXiv200711623B
 Keywords:

 Mathematics  Analysis of PDEs;
 35P30;
 35B09;
 35B32;
 35B34;
 35J62;
 35J20
 EPrint:
 22 pages, 3 figures. Minor textual corrections. Published in Communications in Contemporary Mathematics