A priori estimates, existence and non-existence of positive solutions of generalized mean curvature equations

2011 ◽  
Vol 74 (18) ◽  
pp. 7126-7136 ◽  
Author(s):  
Qiuyi Dai ◽  
Yonggeng Gu ◽  
Jiuyi Zhu
Keyword(s):  
Positive Solutions ◽  
Mean Curvature ◽  
A Priori Estimates ◽  
A Priori ◽  
Priori Estimates ◽  
Existence Of Positive Solutions ◽  
Mean Curvature Equations ◽  
Generalized Mean Curvature ◽  
Generalized Mean

A priori bounds and existence of positive solutions to a p-kirchhoff equations

2021 ◽  
pp. 2150082
Author(s):  
Pengfei Li ◽  
Junhui Xie
Keyword(s):  
Positive Solutions ◽  
Blow Up ◽  
A Priori Estimates ◽  
A Priori ◽  
Auxiliary Problem ◽  
Degree Theory ◽  
Kirchhoff Equations ◽  
Priori Estimates ◽  
Existence Of Positive Solutions ◽  
Dirichlet Boundary Problem

In this paper, we consider a [Formula: see text]-Kirchhoff problem with Dirichlet boundary problem in a bounded domain. Under suitable conditions, we get a priori estimates for positive solutions to an auxiliary problem by the well-known blow-up argument. As an application, a existence result for positive solutions is proved by the topological degree theory.

scholarly journals Positive Solutions for a System of Fourth-Order p-Laplacian Boundary Value Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Lianlong Sun ◽  
Zhilin Yang
Keyword(s):  
Boundary Value Problems ◽  
Positive Solutions ◽  
Fourth Order ◽  
A Priori Estimates ◽  
Boundary Value ◽  
A Priori ◽  
Index Theory ◽  
Priori Estimates ◽  
Fixed Point Index Theory ◽  
Existence Of Positive Solutions

We investigate the existence of positive solutions for the system of fourth-order p-Laplacian boundary value problems (|u′′|p-1u′′)′′=f1(t,u,v),  (|v′′|q-1v′′)′′=f2(t,u,v),  u(2i)(0)=u(2i)(1)=0,  i=0,1,  v(2i)(0)=v(2i)(1)=0,  i=0,1, where p,q>0 and f1,f2∈C([0,1]×ℝ+2,ℝ+)  (ℝ+:=[0,∞)). Based on a priori estimates achieved by utilizing Jensen’s integral inequalities and nonnegative matrices, we use fixed point index theory to establish our main results.

A direct blowing-up and rescaling argument on nonlocal elliptic equations

2016 ◽  
Vol 27 (08) ◽  
pp. 1650064 ◽  
Author(s):  
Wenxiong Chen ◽  
Congming Li ◽  
Yan Li
Keyword(s):  
Positive Solutions ◽  
Fractional Laplacian ◽  
A Priori Estimates ◽  
A Priori ◽  
Degree Theory ◽  
Nonlocal Operator ◽  
Nonlocal Operators ◽  
Priori Estimates ◽  
Blowing Up ◽  
Existence Of Positive Solutions

In this paper, we develop a direct blowing-up and rescaling argument for nonlinear equations involving nonlocal elliptic operators including the fractional Laplacian. Instead of using the conventional extension method introduced by Caffarelli and Silvestre to localize the problem, we work directly on the nonlocal operator. Using the defining integral, by an elementary approach, we carry on a blowing-up and rescaling argument directly on the nonlocal equations and thus obtain a priori estimates on the positive solutions. Based on this estimate and the Leray–Schauder degree theory, we establish the existence of positive solutions. We believe that the ideas introduced here can be applied to problems involving more general nonlocal operators.

scholarly journals Regularity and existence of positive solutions for a fractional system

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Ran Zhuo ◽  
Yan Li
Keyword(s):  
Positive Solutions ◽  
A Priori Estimates ◽  
A Priori ◽  
Degree Theory ◽  
A Priori Estimate ◽  
Technical Difficulty ◽  
Priori Estimates ◽  
Existence Of Positive Solutions ◽  
Regularity And Existence ◽  
Fractional Orders

<p style='text-indent:20px;'>We consider the nonlinear fractional elliptic system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{\begin{array}{ll} (- \Delta)^{\frac{\alpha_1}{2}}u(x) = f(x, u, v), &amp; \text{in}\, \, \, \Omega, \\ (- \Delta)^{\frac{\alpha_2}{2}}v(x) = g(x, u, v), &amp; \text{in}\, \, \, \Omega, \\ u = v = 0, &amp; \text{in}\, \, \, \mathbb{R}^n\setminus\Omega, \end{array} \right. \label{a-1.2} \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ 0&lt;\alpha_1, \alpha_2&lt;2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a bounded domain with <inline-formula><tex-math id="M3">\begin{document}$ C^2 $\end{document}</tex-math></inline-formula> boundary in <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula>. To overcome the technical difficulty due to the different fractional orders, we employ two distinct methods and derive the a priori estimates for <inline-formula><tex-math id="M5">\begin{document}$ 0&lt;\alpha_1, \alpha_2&lt;1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ 1&lt;\alpha_1, \alpha_2 &lt;2 $\end{document}</tex-math></inline-formula> respectively. Moreover, combining the a priori estimate with the topological degree theory, we prove the existence of positive solutions.</p>

scholarly journals Quasilinear elliptic systems in divergence form associated to general nonlinearities

2018 ◽  
Vol 7 (4) ◽  
pp. 425-447 ◽  
Author(s):  
Lorenzo D’Ambrosio ◽  
Enzo Mitidieri
Keyword(s):  
Positive Solutions ◽  
A Priori Estimates ◽  
A Priori ◽  
Elliptic Systems ◽  
Divergence Form ◽  
Systems Of Equations ◽  
Quasilinear Elliptic Systems ◽  
Open Set ◽  
Priori Estimates ◽  
Quasilinear Elliptic

AbstractThe paper is concerned with a priori estimates of positive solutions of quasilinear elliptic systems of equations or inequalities in an open set of {\Omega\subset\mathbb{R}^{N}} associated to general continuous nonlinearities satisfying a local assumption near zero. As a consequence, in the case {\Omega=\mathbb{R}^{N}}, we obtain nonexistence theorems of positive solutions. No hypotheses on the solutions at infinity are assumed.

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