Research on the existence of solutions of differential equations and their properties via functional analysis methods

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For elliptic problems which have Nehari manifold structures, there were some results on the numbers of their positive solutions. However, the numbers of sign-changing solutions were not studied well. By using category on the domain, we give a result on a number of sign-changing solutions. For elliptic problems which have Sobolev's critical exponent, the existence of solutions whose level are greater that the two times of the level of the limit equation were not discussed well. We gave such an argument to show the existence of the multiple solutions.We also studied the uniqueness of positive radial solutions for elliptic equations. The uniqueness of a positive radial solution of the scalar field equation has a long history. We gave a uniqueness result which is applicable to various equations including the scalar field equation.