Existence Of Solution For Resonant Double Phase Problem

In this paper,we study the existence of solutions of the double phase equation at resonance at the first eigenvalue of the relevant characteristic equation.It mainly includes the following two parts.One is the double phase problem with mixed boundary conditions.When the first eigenvalue λ₁（q）of the q-Laplacian characteristic equation with mixed boundary conditions is resonant,the existence and multiplicity of solutions are obtained.Through the local（1,1）linking theory and the three critical point theorem in Morse theory,it is concluded that there are at least two bounded nontrivial solutions for the two-phase equation with mixed boundary conditions at λ₁（q）when resonance occurs.The second is the existence of periodic solutions for one-dimensional double phase periodic equations when the first eigenvalue λ₁（p）of the non-homogeneous p-Laplacian periodic problem resonates.By applying some nonlinear conditions to the reaction term f（x,y）and using the mountain pass theorem,we obtain the result that the one-dimensional double phase periodic equation has at least one periodic solution at resonance at λ₁（p）.Since the resonance condition makes the energy functional of the double phase equation not satisfy the（PS）-condition,the C-condition with weaker conditions is used instead.