Solutions of a system of diffusion equations
(2007) Journal of Fixed Point Theory and Applications — Vol. 2, n° 1, p. 117-139 (2007)
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- Abstract
- We study existence and multiplicity of homoclinic type solutions to the following system of diffusion equations on R x Omega: {partial derivative(t)u - Delta(x)u + b(t, x) . del(x) u + V(x)u = H-v (t, x, u, v), {-partial derivative(t)v - Delta(x)v - b(t, x) . del(x) v + V(x)v = H-u (t, x, u, v), where Omega = R-N or Omega is a smooth bounded domain of R-N, z = (u, v) : R x Omega -> R-m x R-m, and b is an element of C-1 (R x (Omega) over bar, R-N), V is an element of C((Omega) over bar, R), H is an element of C-1 (R x (Omega) over bar x R-2m, R), all three depending periodically on t and x. We assume that H(t, x, 0) 0 and H is asymptotically quadratic or superquadratic as vertical bar z vertical bar -> infinity. The superquadratic condition is more general than the usual one. By establishing a proper variational setting based on some recent critical point theorems we obtain at least one nontrivial solution, and also infinitely many solutions provided H is moreover symmetric in z.
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Ding, Y., Luan, S., & Willem, M. (2007). Solutions of a system of diffusion equations. Journal of Fixed Point Theory and Applications, 2(1), 117-139. https://doi.org/10.1007/s11784-007-0023-8 (Original work published 2007)