Positive solutions for a class of p-Laplacian systems with multiple parameters
by
Jaffar Ali ShahulHameed
Mississippi State University
Coauthors: Ratnasingham Shivaji

Abstract

- D_p u=l₁f(v)+m₁ h(u) in W
-D_q v=l₂g(u)+m₂ g(v) in W
u=0=v on ∂W

where D_s z=div(|∇z|^s-2∇z), s > 1, l₁, l₂, m₁ and m₂ are non-negative parameters, and W is a bounded domain in Rⁿ with smooth boundary ∂W. We prove the existence of a large positive solution for l₁+m₁ and l₂+m₂ large when

lim x→∞  f(M[g(x)]1/q-1) xp-1 =0

for every M > 0, lim_x→∞[h(x)/(x^p-1)]=0 and lim_x→∞[(g(x))/(x^q-1)]=0. In particular, we do not assume any sign conditions on f(0), g(0), h(0) or g(0). We also discuss a multiplicity results when f(0)=g(0)=h(0)=g(0)=0.

Date received: October 1, 2007