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

We study the zero Dirichlet problem for equation $-\Delta_p u -\Delta_q = \alpha |u|^{p-2}u+\beta |u|^{q-2}u$ in a bounded domain $\Omega \subset \mathbb{R}^N$, with $1<q<p$. investigate relation between two critical curves on $(\alpha,\beta)$-plane corresponding to threshold of existence special classes positive solutions. In particular, certain neighbourhoods 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 first eigenfunction $p$-Laplacian, we show and, which rather unexpected, three distinct solutions, depending exponents $p$ and $q$.