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Nonexistence results of sign-changing solutions to a supercritical nonlinear problem

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  • In this paper we study the nonlinear elliptic problem involving nearly critical exponent $ (P_\varepsilon ): -\Delta u= $ $ |u|^{4/(n-2)+\varepsilon}u$ in $\Omega$, $u = 0$ on $\partial \Omega $, where $\Omega $ is a smooth bounded domain in $\mathbb R^n $, $n \geq 3 $ and $\varepsilon$ is a positive real parameter. We show that, for $\varepsilon$ small, $(P_\varepsilon) $ has no sign-changing solutions with low energy which blow up at two points. Moreover, we prove that there is no sign-changing solutions which blow up at three points. We also show that $(P_\varepsilon)$ has no bubble-tower sign-changing solutions.
    Mathematics Subject Classification: Primary: 35J20; Secondary: 35J60.


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