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MULTIPLE SOLUTIONS FOR A KIRCHHOFF EQUATION WITH NONLINEARITY HAVING ARBITRARY GROWTH
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 96 / Issue 1 / August 2017
- Published online by Cambridge University Press:
- 13 March 2017, pp. 98-109
- Print publication:
- August 2017
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- Article
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- You have access
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We prove the existence of infinitely many solutions
$u\in W_{0}^{1,2}(\unicode[STIX]{x1D6FA})$ for the Kirchhoff equation where
$$\begin{eqnarray}\displaystyle -\biggl(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD}\int _{\unicode[STIX]{x1D6FA}}|\unicode[STIX]{x1D6FB}u|^{2}\,dx\biggr)\unicode[STIX]{x1D6E5}u=a(x)|u|^{q-1}u+\unicode[STIX]{x1D707}f(x,u)\quad \text{in }\unicode[STIX]{x1D6FA}, & & \displaystyle \nonumber\end{eqnarray}$$
$\unicode[STIX]{x1D6FA}\subset \mathbb{R}^{N}$ is a bounded smooth domain, 
$a(x)$ is a (possibly) sign-changing potential, 
$0<q<1$, 
$\unicode[STIX]{x1D6FC}>0$, 
$\unicode[STIX]{x1D6FD}\geq 0$, 
$\unicode[STIX]{x1D707}>0$ and the function 
$f$ has arbitrary growth at infinity. In the proof, we apply variational methods together with a truncation argument.
