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Stationary Solutions and Connecting Orbits for <Emphasis Type="Italic">p</Emphasis>-Laplace Equation
Authors:Aleksander ?wiszewski  Mateusz Maciejewski
Institution:1.Faculty of Mathematics and Computer Science,Nicolaus Copernicus University,Toruń,Poland
Abstract:
We deal with one dimensional p-Laplace equation of the form
$$\begin{aligned} u_t = (|u_x|^{p-2} u_x )_x + f(x,u), \ x\in (0,l), \ t>0, \end{aligned}$$
under Dirichlet boundary condition, where \(p>2\) and \(f:0,l]\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is a continuous function with \(f(x,0)=0\). We will prove that if there is at least one eigenvalue of the p-Laplace operator between \(\lim _{u\rightarrow 0} f(x,u)/|u|^{p-2}u\) and \(\lim _{|u|\rightarrow +\infty } f(x,u)/|u|^{p-2}u\), then there exists a nontrivial stationary solution. Moreover we show the existence of a connecting orbit between stationary solutions. The results are based on Conley index and detect stationary states even when those based on fixed point theory do not apply. In order to compute the Conley index for nonlinear semiflows deformation along p is used.
Keywords:
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