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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
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 $u=0$.
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- © The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.