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MULTIPLE SOLUTIONS FOR
$p(x)$-LAPLACIAN EQUATIONS WITH NONLINEARITY SUBLINEAR AT ZERO
Published online by Cambridge University Press: 29 January 2024
Abstract
We consider the Dirichlet problem for $p(x)$-Laplacian equations of the form
$$ \begin{align*} -\Delta_{p(x)}u+b(x)\vert u\vert ^{p(x)-2}u=f(x,u),\quad u\in W_{0}^{1,p(x)}(\Omega). \end{align*} $$
The odd nonlinearity $f(x,u)$ is
$p(x)$-sublinear at
$u=0$ but the related limit need not be uniform for
$x\in \Omega $. Except being subcritical, no additional assumption is imposed on
$f(x,u)$ for
$|u|$ large. By applying Clark’s theorem and a truncation method, we obtain a sequence of solutions with negative energy and approaching the zero function
$u=0$.
MSC classification
- Type
- Research Article
- Information
- Copyright
- © The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
References
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