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arXiv · 1411.5192

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

Vladimir Bobkov·Mieko Tanaka
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Abstract

We study the existence and non-existence of positive solutions for the $(p,q)$-Laplace equation $-\Delta_p u -\Delta_q u = \alpha |u|^{p-2} u + \beta |u|^{q-2} u$, where $p \neq q$, under the zero Dirichlet boundary condition in $\Omega$. The main result of our research is the construction of a continuous curve in $(\alpha,\beta)$ plane, which becomes a threshold between the existence and non-existence of positive solutions. Furthermore, we provide the example of domains $\Omega$ for which the corresponding first Dirichlet eigenvalue of $-\Delta_p$ is not monotone w.r.t. $p > 1$.

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Vladimir Bobkov, Mieko Tanaka. 2014-11-19. On positive solutions for $(p,q)$-Laplace equations with two parameters. https://doi.org/10.1007/s00526-015-0903-5

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