By variational methods and the Moser iteration techniques, the authors proved the existence of positive solutions and some properties of the nontrivial solutions to (1.3).
By variational methods and the symmetric criticality principle of Palais, we establish several existence and multiplicity results of positive G-symmetric solutions under certain appropriate hypotheses on Q, h, and q.
When (mathbb{T}=mathbb{R}), (I_{ij}equiv0), (iinLambda_{1}), (jin Lambda_{2}), B and (A t)) are not zero matrices, Li et al. in [7] researched the existence and multiplicity of solutions for (1) by variational methods and some critical point theorems.
When the elastic beam equation does not contain parameter λ, the existence of multiple positive solutions and unique positive solution was presented in [17] by variational methods and in [18] by a fixed point theorem, respectively; monotone positive solutions were obtained by using the monotone iteration method in [19].
In fact, it is a challenging research to study the concentration problem of solutions for the fractional Schrödinger equations by variational methods and it is not trivial to construct the fractional Sobolev space, which is used to study the concentration phenomenon.
We get the first result by a variational method and critical point theory, and we get the second result by homology theory.
end{cases} (1.2) Comte and Knaap [8] proved that there exists a nontrivial solution of problem (1.2) by variational method if (Q x)=1) and (lambda =-mu ).
The authors proved these results by the variational method and the mountain pass theorem.
The optimal functions of the temperature distributions of the automobile exhaust pipe and TPVC are obtained by the variational method and the modified Lagrangian formulation.
