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MULTIPLE SOLUTIONS FOR
$p(x)$-LAPLACIAN EQUATIONS WITH NONLINEARITY SUBLINEAR AT ZERO
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 110 / Issue 2 / October 2024
- Published online by Cambridge University Press:
- 29 January 2024, pp. 346-354
- Print publication:
- October 2024
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- Article
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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 
$u=0$.
