Stability Results for a Class of Elliptic Problems
G. A. Afrouzi, Zahra Sadeghi · 2008
We prove stability/instability of positive stationary solutions to certain classes of elliptic problems of the form ‰ i4'(u(x)) = ‚a(x)f(u(x)) x 2 ›; Bfiu(x) = 0 x 2 @›; where 4 denotes the Laplacian operator, › is a bounded and regular domain in R n ;(n ‚ 1) having smooth boundary Bfiu(x) = fih(x)u(x) + (1 i fi) @u @n where fi 2 (0;1) is a constant and h : @› ! R + is a smooth function with h · 1 when fi = 1, ‚ > 0, and '(u) 2 R + and f are smooth functions with '(0) = 0. We also assume that weight a(x) : › ! R satisfies either a(x) > 0 or a(x) 0 is a parameter, › is a bounded and regular domain in R n having smooth boundary Bfiu(x) = fih(x)u(x) + (1 i fi) @u where fi 2 (0;1) is a constant and h : @› ! R + is a smooth function with h · 1 when fi = 1, i.e; the boundary condition may be of Dirichlet(u = 0), Neumann ( @u = 0) or mixed type (robin boundary condition), and '(u) 2 R + is a smooth function with '(0) = 0, and weight function a(x) : › ! R satisfies either a(x) > 0 or a(x) < 0 for all x 2 ›. We shall assume throughout that smooth function f satisfies either of the hypotheses (F) or (G) below: