Nonexistence of Sign-Changing Solutions for Some Elliptic Inequalities in Bounded Domains

2020 ◽  
Vol 60 (8) ◽  
pp. 1331-1336
Author(s):  
E. I. Galakhov ◽  
O. A. Salieva
Keyword(s):  
Bounded Domains ◽  
Sign Changing Solutions ◽  
Elliptic Inequalities

On a Kirchhoff Equation in Bounded Domains

2018 ◽  
Vol 18 (3) ◽  
pp. 613-648
Author(s):  
Yisheng Huang ◽  
Yuanze Wu
Keyword(s):  
Variational Method ◽  
Smooth Boundary ◽  
Uniqueness Result ◽  
Kirchhoff Equation ◽  
Bounded Domains ◽  
Multiplicity Results ◽  
Nodal Domains ◽  
Existence And Multiplicity ◽  
Sobolev Exponent ◽  
Sign Changing Solutions

AbstractIn this paper, we consider the following Kirchhoff equation:\left\{\begin{aligned} &\displaystyle{-}\bigg{(}a+b\int_{\Omega}\lvert\nabla u% |^{2}\,dx\bigg{)}\Delta u=\lambda u+|u|^{p-2}u&&\displaystyle\text{in }\Omega,% \\ &\displaystyle u=0&&\displaystyle\text{on }\partial\Omega,\end{aligned}\right.where{\Omega\subset\mathbb{R}^{N}}({N\geq 3}) is a bounded domain with smooth boundary{\partial\Omega},{2<p<2^{*}=\frac{2N}{N-2}}is the Sobolev exponent anda,b, λ are positive parameters. By the variational method, we obtain some existence and multiplicity results of the sign-changing solutions (including the radial sign-changing solution in the case of{\Omega=\mathbb{B}_{R}}) for this problem. Some further properties of these sign-changing solutions, such as the numbers of the nodal domains, the concentration behaviors as{b\to 0^{+}}, the estimates of the energy values and so on, are also obtained. Our results generalize and improve some known results in the literature. Moreover, we also obtain a uniqueness result of the radial positive solution.

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