Abstract
We extend to the vector $p-$Laplace operator $(\phi_p(u'))'=(|u'|^{p-2}u')',$ $p>1,$ ($|\cdot|$ denotes the Euclidean norm in $\mathbb R^N$) a method that uses a suitable homotopy along $p$ to evaluate at the level $p=2$ a Leray-Schauder degree for an associated operator depending on $p.$ We apply this result to prove existence of nontrivial solutions to the system $$(\phi_p(u'))'=f(t,u)\quad\text{a.e.}\quad \mbox{in}\quad (0,T),\quad u(0)=0,\quad u(T)=0, $$ where $f:[0,T]\times\mathbb R^N\mapsto \mathbb R^N$ is a Carathéodory function, with $f(t,0)=0.$
Citation
M. García-Huidobro. R. Manásevich. J. R. Ward. "A homotopy along $p$ for systems with a vector $p$-Laplace operator." Adv. Differential Equations 8 (3) 337 - 356, 2003. https://doi.org/10.57262/ade/1355926857
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