Hardy–Sobolev critical singular elliptic equations with mixed Dirichlet–Neumann boundary conditions

Three important inequalities (the Poincaré, Hardy and generalized Poincaré inequalities) on the mixed boundary conditions are firstly established by some analytical techniques. Then the existence and multiplicity of positive solutions are studied for a class of semilinear elliptic equations with mixed Dirichlet–Neumann boundary conditions involving Hardy terms and Hardy–Sobolev critical exponents by using the variational methods.