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.

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 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.

The Dirichlet problem for the uniformly elliptic equation in generalized weighted Morrey spaces

2020 ◽  
Vol 57 (1) ◽  
pp. 68-90 ◽  
Author(s):  
Tahir S. Gadjiev ◽  
Vagif S. Guliyev ◽  
Konul G. Suleymanova
Keyword(s):  
Elliptic Equations ◽  
A Priori Estimates ◽  
A Priori ◽  
Morrey Spaces ◽  
Smooth Bounded Domain ◽  
Weighted Morrey Spaces ◽  
Priori Estimates ◽  
Muckenhoupt Class ◽  
Morrey Estimates ◽  
Dirichlet Boundary Problem

Abstract In this paper, we obtain generalized weighted Sobolev-Morrey estimates with weights from the Muckenhoupt class Ap by establishing boundedness of several important operators in harmonic analysis such as Hardy-Littlewood operators and Calderon-Zygmund singular integral operators in generalized weighted Morrey spaces. As a consequence, a priori estimates for the weak solutions Dirichlet boundary problem uniformly elliptic equations of higher order in generalized weighted Sobolev-Morrey spaces in a smooth bounded domain Ω ⊂ ℝn are obtained.

Sign in / Sign up

Export Citation Format

Share Document