Picone’s identity for -Laplace operator and its applications

UDC 517.9We prove a nonlinear analogue of Picone's identity for -Laplace operator. As an application, we give a Hardy type inequality and Sturmian comparison principle.We also show the strict monotonicity of the principle eigenvalue and degenerate elliptic system.

Liouville type theorems for stable solutions of elliptic system involving the Grushin operator

<p style='text-indent:20px;'>We examine the following degenerate elliptic system:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ -\Delta_{s} u \! = \! v^p, \quad -\Delta_{s} v\! = \! u^\theta, \;\; u, v&gt;0 \;\;\mbox{in }\; \mathbb{R}^N = \mathbb{R}^{N_1}\times \mathbb{R}^{N_2}, \quad\mbox{where}\;\; s \geq 0\;\; \mbox{and} \;\;p, \theta \!&gt;\!0. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>We prove that the system has no stable solution provided <inline-formula><tex-math id="M1">\begin{document}$ p, \theta &gt;0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ N_s: = N_1+(1+s)N_2&lt; 2 + \alpha + \beta, $\end{document}</tex-math></inline-formula> where</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \alpha = \frac{2(p+1)}{p\theta - 1} \quad\mbox{and} \quad \beta = \frac{2(\theta +1)}{p\theta - 1}. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>This result is an extension of some results in [<xref ref-type="bibr" rid="b15">15</xref>]. In particular, we establish a new integral estimate for <inline-formula><tex-math id="M3">\begin{document}$ u $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ v $\end{document}</tex-math></inline-formula> (see Proposition 1.1), which is crucial to deal with the case <inline-formula><tex-math id="M5">\begin{document}$ 0 &lt; p &lt; 1. $\end{document}</tex-math></inline-formula></p>

Liouville-type theorem for high order degenerate Lane-Emden system

<p style='text-indent:20px;'>In this paper, we are concerned with the following high order degenerate elliptic system:</p><p style='text-indent:20px;'><disp-formula><label/><tex-math id="FE111000">\begin{document}$\left\{ \begin{align} &amp; {{(-A)}^{m}}u={{v}^{p}} \\ &amp; {{(-A)}^{m}}v={{u}^{q}}\quad \text{ in }\mathbb{R}_{+}^{n+1}:=\left\{ (x,y)|x\in {{\mathbb{R}}^{n}},y&gt;0 \right\}, \\ &amp; u\ge 0,v\ge 0 \\ \end{align} \right.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( 1 \right)$\end{document}</tex-math></disp-formula></p><p style='text-indent:20px;'>where the operator <inline-formula><tex-math id="M1">\begin{document}$ A: = y\partial_{y}^2+a\partial_{y}+\Delta_{x}, \;a\geq 1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ n+2a&gt;2m, m\in \mathbb{Z}^+,\;p,\,q\geq 1 $\end{document}</tex-math></inline-formula>. We prove the non-existence of positive smooth solutions for <inline-formula><tex-math id="M3">\begin{document}$ 1&lt;p,\, q&lt;\frac{n+2a+2m}{n+2a-2m} $\end{document}</tex-math></inline-formula>, and classify positive solutions for <inline-formula><tex-math id="M4">\begin{document}$ p = q = \frac{n+2a+2m}{n+2a-2m} $\end{document}</tex-math></inline-formula>. For <inline-formula><tex-math id="M5">\begin{document}$ \frac{1}{p+1}+\frac{1}{q+1}&gt;\frac{n+2a-2m}{n+2a} $\end{document}</tex-math></inline-formula>, we show the non-existence of positive, ellipse-radial, smooth solutions. Moreover we prove the non-existence of positive smooth solutions for the high order degenerate elliptic system of inequalities <inline-formula><tex-math id="M6">\begin{document}$ (-A)^{m}u\geq v^p, (-A)^{m}v\geq u^q, u\geq 0, v\geq 0, $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M7">\begin{document}$ \mathbb{R}_+^{n+1} $\end{document}</tex-math></inline-formula> for either <inline-formula><tex-math id="M8">\begin{document}$ (n+2a-2m)q&lt;\frac{n+2a}{p}+2m $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M9">\begin{document}$ (n+2a-2m)p&lt;\frac{n+2a}{q}+2m $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M10">\begin{document}$ p,q&gt;1 $\end{document}</tex-math></inline-formula>.</p>

Existence and non-existence of solutions to a Hamiltonian strongly degenerate elliptic system

We study the non-existence and existence of infinitely many solutions to the semilinear degenerate elliptic system \left\{\begin{aligned} &\displaystyle{-}\Delta_{\lambda}u=\lvert v\rvert^{p-1}% v&&\displaystyle\phantom{}\text{in }\Omega,\\ &\displaystyle{-}\Delta_{\lambda}v=\lvert u\rvert^{q-1}u&&\displaystyle% \phantom{}\text{in }\Omega,\\ &\displaystyle u=v=0&&\displaystyle\phantom{}\text{on }\partial\Omega,\end{% aligned}\right. in a bounded domain {\Omega\subset\mathbb{R}^{N}} with smooth boundary {\partial\Omega} . Here {p,q>1} , and {\Delta_{\lambda}} is the strongly degenerate operator of the form \Delta_{\lambda}u=\sum^{N}_{j=1}\frac{\partial}{\partial x_{j}}\Bigl{(}\lambda% _{j}^{2}(x)\frac{\partial u}{\partial x_{j}}\Bigr{)}, where {\lambda(x)=(\lambda_{1}(x),\dots,\lambda_{N}(x))} satisfies certain conditions.

The Neumann Problem for a Degenerate Elliptic System Near Resonance

This paper studies the following system of degenerate equations-divpx∇u+qxu=αu+βv+g1x,v+h1x,x∈Ω,-div(p(x)∇v)+q(x)v=βu+αv+g2(x,u)+h2(x),x∈Ω,∂u/∂ν=∂v/∂ν=0,x∈∂Ω.HereΩ⊂Rnis a boundedC2domain, andνis the exterior normal vector on∂Ω. The coefficient functionpmay vanish inΩ¯,q∈Lr(Ω)withr>ns/(2s-n), s>n/2. We show that the eigenvalues of the operator-div(p(x)∇u)+q(x)uare discrete. Secondly, when the linear part is near resonance, we prove the existence of at least two different solutions for the above degenerate system, under suitable conditions onh1,h2,g1, andg2.