The Positive Solutions Of Two Semi-linear Elliptic Equations With Different Boundary Conditions

In this paper ,we study two problems:1, We concern with positive solutions of the degenerate logistic type semi-linear equationfor T = {x =(x₁,x₂,…,x_N):x_N≥0}, (N≥2), a(x) and b(x) are continuous functions on T (?) R^N and b(x) satisfies b(x)≥0, b(x) (?) 0; b(x)≡0, x∈Ω₀.Ω₀ is bounded connected set forΩ₀ = {x∈Ω₀: b(x)= 0}, (?)Ω₀ is smooth. n is outward pointing unit normal vector of (?)T.The case of "x in bounded region"is widely treated in many literatures (for example [8]).Here, by using sub-super solution and a new approach,we show that there exists a unique positive solution on a unbounded region.2, We concern with positive solutions of semi-linear equationwhereφ(x) is continuous function forφ(x)≥C₀, C₀ andσare positive constants, f(u) is locally quasi-monotone on [0,∞)and satisfies f(u) > 0, u∈(0, a); f(u) < 0, u∈(a,∞).Furthermore we study the following semi-linear equationwhere f(x,u) satisfiesφ₁(x)f₁(u)≤f(x,u)≤φ₂(x)f₂{u),which f₁ satisfies f₁(u) > 0, u∈(0,a₁), f₁(u) < 0, u∈(a₁,∞) and f₂ satisfies f₂(u) > 0, u∈(0,a₂), f₂(u) < 0, u∈(a₂,∞). 0 < C₁≤h(x)≤C₂, C_i, i = 1,2 are positive constants and f_i(u) are locally quasi-monotone functions on [0,∞).In[2]or[3],the right of the equation must be f(u) and the boundary condition is "u =α" whereαis nonnegative constants or∞.Here,we study the case of the non-autonomous functions(rely on x) on the right equations and bounded variant boundary conditions. Obviously, the strong maximum principle makes no sense under our assumptions,but we can use a new approachbased on the weak sweeping principle to show that the equation exists a unique bounded positive solution . Our results are improved the corresponding ones in [3].