Existence and Multiplicity of Solutions for a Class of Particular Boundary Value Poisson Equations - Songyue Yu; Baoqiang Yan (Mathematics, 2022)
In this paper, a special class of boundary value problems, <semantics>−▵u=λuq+ur,in<mspace width="4pt"></mspace>Ω,</semantics><semantics>u>0</semantics>, <semantics>in<mspace width="4pt"></mspace>Ω,</semantics><semantics>n·∇u+g(u)u=0,on<mspace width="4pt"></mspace>∂Ω,</semantics> where <semantics><mstyle scriptlevel="0" displaystyle="true">0<q<1<r<N+2N−2</mstyle></semantics> and <semantics>g:[0,∞)→(0,∞)</semantics> is a nondecreasing <semantics>C1</semantics> function. Here, <semantics>Ω⊂RN</semantics>(<semantics>N≥3</semantics>) is a bounded domain with smooth boundary <semantics>∂Ω</semantics> and <semantics>λ>0</semantics> is a parameter. The existence of the solution is verified via sub- and super-solutions method. In addition, the influences of parameters on the minimum solution are also discussed. The second positive solution is obtained by using the variational method.