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

Abstract

We study the zero Dirichlet problem for the equation $-Δ_p u -Δ_q u = α|u|^{p-2}u+β|u|^{q-2}u$ in a bounded domain $Ω\subset \mathbb{R}^N$, with $1<q<p$. We investigate the relation between two critical curves on the $(α,β)$-plane corresponding to the threshold of existence of special classes of positive solutions. In particular, in certain neighbourhoods of the point $(α,β) = \left(\|\nabla φ_p\|_p^p/\|φ_p\|_p^p, \|\nabla φ_p\|_q^q/\|φ_p\|_q^q\right)$, where $φ_p$ 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 $q$.