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arXiv · 2007.11623

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

Vladimir Bobkov·Mieko Tanaka
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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 $(\alpha,\beta) = \left(\|\nabla \varphi_p\|_p^p/\|\varphi_p\|_p^p, \|\nabla \varphi_p\|_q^q/\|\varphi_p\|_q^q\right)$, 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 $q$.

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Vladimir Bobkov, Mieko Tanaka. 2020-07-22. Multiplicity of positive solutions for $(p,q)$-Laplace equations with two parameters. https://doi.org/10.1142/s0219199721500085

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