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Positive steady--states for two interacting species models with linear self-cross diffusions
On positive solutions for classes of p-Laplacian semipositone systems
| 1. | Department of Mathematical Sciences, The University of North Carolina at Greensboro, Greensboro, NC 27402, United States |
| 2. | Department of Mathematics and Statitstics, Mississippi State University, Mississippi State, MS 39762, United States, United States |
$-\Delta_p u = \lambda f(v)$ in $\quad \Omega $
$-\Delta_p v = \lambda g(u)$ in $ \quad \Omega $
$u = 0 = v$ on $ \quad \partial \Omega$
where $ \lambda > 0 $ is a parameter, $ \Delta_p $ denotes the p-Laplacian operator defined by $ \Delta_p(z)$:=div$(|\nabla z|^{p-2}\nabla z) $ for $ p> 1 $ and $ \Omega $ is a bounded domain with smooth boundary. Here $ f,g \in C[0,\infty) $ belong to a class of functions satisfying $ \lim_{z \to \infty}\frac{f(z)}{z^{p-1}}=0, \lim_{z \to \infty}\frac{g(z)}{z^{p-1}}=0 $. In particular, we discuss the existence of radial solutions for large $ \lambda $ when $ \Omega $ is an annulus. For a general bounded region $ \Omega, $ we also discuss a non-existence result when $ f(0) < 0 $ and $ g(0) < 0. $
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