Positive radial solutions for quasilinear equations in the annulus

Pages: 878 - 885, Issue Special, August 2005

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Haiyan Wang - Division of Mathematical and Natural Sciences, Arizona State University, Phoenix, AZ 85069-7100, United States (email)

Abstract: The paper deals with the existence of positive radial solutions for the quasilinear system $\textrm{ div} \left ( | \nabla u_i|^{p-2}\nabla u_i \right ) + f^i(u_1,...,u_n)=0,\; p>1, R_1 <|x| < R_2,\;u_i(x)=0,$ on $|x|=R_1$ and $R_2$, $i=1,...,n$, $x \in \mathbb{R}^N.$ $f^i$, $i=1,...,n,$ are continuous and nonnegative functions. Let $\vect{u}=(u_1,...,u_n),$ $\varphi(t)=|t|^{p-2}t,$ $f_0^i =\lim_{\norm{\vect{u}} \to 0} \frac{f^i(\vect{u})}{\var(\norm{\vect{u}})},$ $f_{\infty}^i =\lim_{\norm{\vect{u}} \to \infty} \frac{f^i(\vect{u})}{\var(\norm{\vect{u}})}$, $i=1,...,n,$ $\vect{f}=(f^1,...,f^n),$ $\vect{f}_0=\sum_{i=1}^n f_0^i$ and $\vect{f}_{\infty}=\sum_{i=1}^n f_{\infty}^i$. We prove that $\vect{f}_0 =0$ and $\vect{f}_{\infty}=\infty$ (superlinear) guarantee the existence of positive radial solutions for the system. We shall use fixed point theorems in a cone.

Keywords:  p-Laplacian, fixed index theorem, positive radial solution.
Mathematics Subject Classification:  Primary: 35J65 Secondary: 34B15.

Received: September 2004;      Revised: May 2005;      Published: September 2005.