scholarly journals On the number of critical points of solutions of semilinear elliptic equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Massimo Grossi

<p style='text-indent:20px;'>In this survey we discuss old and new results on the number of critical points of solutions of the problem</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE0.1"> \begin{document}$ \begin{equation} \begin{cases} -\Delta u = f(u)&amp;in\ \Omega\\ u = 0&amp;on\ \partial \Omega \end{cases} \;\;\;\;\;\;\;\;(0.1)\end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset \mathbb{R}^N $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M2">\begin{document}$ N\ge2 $\end{document}</tex-math></inline-formula> is a smooth bounded domain. Both cases where <inline-formula><tex-math id="M3">\begin{document}$ u $\end{document}</tex-math></inline-formula> is a positive or nodal solution will be considered.</p>

2014 ◽  
Vol 14 (4) ◽  
Author(s):  
Asadollah Aghajani ◽  
Alireza Mosleh Tehrani ◽  
Nassif Ghoussoub

AbstractWe consider the semilinear elliptic problem −Δu = f (x, u), posed in a smooth bounded domain Ω of ℝ


2021 ◽  
Vol 7 (3) ◽  
pp. 4199-4210
Author(s):  
CaiDan LaMao ◽  
◽  
Shuibo Huang ◽  
Qiaoyu Tian ◽  
Canyun Huang ◽  
...  

<abstract><p>In this paper, we study the summability of solutions to the following semilinear elliptic equations involving mixed local and nonlocal operators</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left\{ \begin{matrix} - \Delta u(x)+{{(-\Delta )}^{s}}u(x)=f(x), &amp; x\in \Omega , \\ u(x)\ge 0,~~~~~ &amp; x\in \Omega , \\ u(x)=0,~~~~~ &amp; x\in {{\mathbb{R}}^{N}}\setminus \Omega , \\ \end{matrix} \right. $\end{document} </tex-math></disp-formula></p> <p>where $ 0 &lt; s &lt; 1 $, $ \Omega\subset \mathbb{R}^N $ is a smooth bounded domain, $ (-\Delta)^s $ is the fractional Laplace operator, $ f $ is a measurable function.</p></abstract>


2014 ◽  
Vol 14 (2) ◽  
Author(s):  
Sara Barile ◽  
Giovany M. Figueiredo

AbstractIn this paper we prove an existence result for a least energy nodal (or sign-changing) solution for the class of p&q problems given bywhere Ω is a smooth bounded domain in ℝ


Author(s):  
Qiuyi Dai ◽  
Yonggeng Gu

Let Ω ⊂ RN be a bounded domain. We consider the nonlinear problem and prove that the existence of positive solutions of the above nonlinear problem is closely related to the existence of non-negative solutions of the following linear problem: .In particular, if p > (N + 2)/(N − 2), then the existence of positive solutions of nonlinear problem is equivalent to the existence of non-negative solutions of the linear problem (for more details, we refer to theorems 1.2 and 1.3 in § 1 of this paper).


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