Study On Existence Of Solutions For Several Kinds Of Double Phase Problems

The double phase equations are a class of nonlinear elliptic differential equations.It is widely used in image processing,non-Newtonian percolation problems,biophysics,plasma physics,elasticity,population dynamics,chemical reaction design and other fields.This paper uses variational methods to study the existence of the solutions to double phase problems,and the main contents are as follows:Firstly,we review the development background and research status of variational methods and double phase problems,and give the required preliminary knowledge.Then,we study the double phase Kirchhoff type equations with critical exponents in the whole space RN Using penalty techniques,truncation methods and Ljusternik-Schnirelmann theory,we prove that when ε>0 is sufficiently small,there are multiple positive solutions to the problem and when ε→ 0,results of concentration for these solutions are obtained.Furthermore,we consider the double phase problems in bounded domains When λ=1,we assume that the nonlinear term f satisfies the asymptotic k condition,and does not satisfy(AR)condition and monotonicity condition.Using the Mountain Pass Theorem we can prove that the problem has at least one non-trivial weak solution.When V(x)=0,λ>0 is a parameter,using the truncation technique and the Mountain Pass Theorem,it is proved that there is at least one positive solution,one negative solution and one signchanging solution to the above problem under the condition that(AR)is not satisfied.In addition,it is proved that when λ is sufficiently large,there are infinitely many sign-changing solutions to the problem.For the double phase problem in RN-div(|▽u|p-2▽u+a(x)|▽u|q-2▽u)+V(x)|u|γ-2u=f(x,u),x∈RN,when the potential function V satisfies certain conditions,and the nonlinear term f satisfies the symmetry condition,the Mountain Pass Theorem and the Fountain Theorem are used to prove the the existence of nontrivial solutions and infinitely many solutions to the above problem.In addition,we deal with the anisotropic double phase problems as follows Using the Mountain Pass Theorem,the existence result of weak solutions to the above problem is proved under certain conditions.In addition,by applying the Fountain Theorem,the Dual Fountain Theorem and the genus theory of Krasnoselskii,the existence of infinitely many solutions is proved.Finally,we pose some interesting questions for further exploration.