Existence And Multiplicity Of Solutions For Elliptic Equations With Nonlinear Boundary Condition

In this paper, we mainly consider semilinear elliptic equations with nonlinear boundary condition. Existence and multiplicity of solutions are studied by the variational methodsand some analysis techniques. Firstly, we consider the following semilinear elliptic systemswhereΩis a bounded domain in R^N with smooth boundary (?) and the parameterλ∈(0,∞). We define R₂⁺ = [0,∞)×[0.∞). F : R₂⁺→R and G : R₂⁺→R satisfy: (g₁) F, G∈C¹(R₂⁺); F(y, z)≥0 and G(y, z)≥0 for all (y, z)∈R₂^;+ F(y, z) (?) 0 andG(y,z)(?)0.(g₂) There exist 2 <α< 2^* = (?) and 1 <β< 2 such that F(ty,tz) = t^αF(y, z) andG(ty,tz) = t^βG(y,z) for all t≥0 and (y,z)∈R₂⁺.(g₃) F_y(0,z) = 0 and G_y(0,z) = 0 for all z∈[0,∞); F_z(y,0) = 0 and G_z(y,0) =0 for all y∈[0,∞).(9₄) There exists a constant M^* such that F_y(y,z)≥M^*y a,nd F_z(y,z)≥M^*z for all(y.,z)∈R₂⁺.We have the following theorem:Theorem 1 Suppose that (g₁) - (g₄) hold. There exists a constantλ^* > 0 such that, for 0 <λ< (?), system (1) has at least two positive solutions. Secondly, the following singular semilinear elliptic problem was studied:where 0∈Q, (?) R^N is a bounded domain with smooth boundary, 0≤μ＜(?) = (?) , 1≤q<2* = (?),λ> 0, v denotes the unit outward normal vectorto boundary (?). The main results of this paper are following theorems:Theorem 2. Assume that l*,0≤μ<μ^*. There existsλ₀ > 0such that for any 0≤λ<λ₀ then problem (2) has a sequence of solutions (?),such that I(u_k) < 0, I(u_k)→0 as k→∞.Theorem 3. Assume that l*,0≤μ<μ^*. There existsλ₀ > 0such that for any 0≤λ<λ₀ then problem (2) has a sequence of solutions (?)such that I(u_k) > 0, I(u_k)→∞as k→∞.