scholarly journals Multiple positive solutions for a fractional elliptic system with critical nonlinearities

2016 ◽  
Vol 2016 (1) ◽  
Author(s):  
Haining Fan
Keyword(s):  
Elliptic System ◽  
Positive Solutions ◽  
Multiple Positive Solutions ◽  
Critical Nonlinearities

Multiple positive solutions for a critical elliptic system with concave—convex nonlinearities

2009 ◽  
Vol 139 (6) ◽  
pp. 1163-1177 ◽  
Author(s):  
Tsing-San Hsu ◽  
Huei-Li Lin
Keyword(s):  
Variational Methods ◽  
Elliptic System ◽  
Positive Solutions ◽  
Critical Growth ◽  
Bounded Domains ◽  
Multiple Positive Solutions ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic System

We consider a semilinear elliptic system with both concave—convex nonlinearities and critical growth terms in bounded domains. The existence and multiplicity results of positive solutions are obtained by variational methods.

scholarly journals Existence of multiple positive solutions for singular p-q-Laplacian problems with critical nonlinearities

2021 ◽  
Author(s):  
Wang Jiayu ◽  
Wei Han
Keyword(s):  
Weak Solution ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Positive Constant ◽  
Smooth Boundary ◽  
Multiple Positive Solutions ◽  
Critical Nonlinearity ◽  
Critical Nonlinearities ◽  
Existence And Multiplicity ◽  
Alpha Beta

In this article, we consider the following p-q-Laplacian system with singular and critical nonlinearity \begin{equation*} \left \{ \begin{array}{lllll} -\Delta_{p}u-\Delta_{q}u=\frac{h_{1}(x)}{u^{r}}+\lambda\frac{\alpha}{\alpha+\beta}u^{\alpha-1}v^{\beta} \ \ in\ \Omega ,\\ -\Delta_{p}v-\Delta_{q}v=\frac{h_{2}(x)}{v^{r}}+\lambda\frac{\beta}{\alpha+\beta}u^{\alpha}v^{\beta-1} \ \ in\ \Omega, \\ u,v>0 \ \ \ \ \ \ in \ \Omega, \ \ \ \ \ u=v=0 \ \ \ \ \ \ \ on \ \partial\Omega, \end{array} \right. \end{equation*} where Ω is a bounded domain in $\mathbb {R}^{n}$ with smooth boundary $\partial\Omega$. $11,\lambda\in(0,\Lambda_{*})$ is parameter with $\Lambda _{*}$ is a positive constant and $h_{1}(x),h_{2}(x)\in L^{\infty},h_{1}(x),h_{2}(x)>0$. We show the existence and multiplicity of weak solution of equation above for suitable range of $\lambda$.

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