Multiplicity of positive solutions for $(p,q)$-Laplace equations with two parameters

Abstract

We study the zero Dirichlet problem for the equation $-{\Delta }_{p}u-{\Delta }_{q}u=\alpha |u{|}^{p-2}u+\beta |u{|}^{q-2}u$ 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 $(\alpha ,\beta )=\left(\Vert \nabla {\phi }_p{\Vert }_p^p/\Vert {\phi }_p{\Vert }_p^p,\Vert \nabla {\phi }_p{\Vert }_q^q/\Vert {\phi }_p{\Vert }_q^q\right)$, where ${\phi }_p$ 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 $q$.

22 pages, 3 figures. Minor textual corrections. Published in Communications in Contemporary Mathematics

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