The Relationship Between The Existence Of The Non-Trivial Solutions And The Volume Of Normal Materials For Ginzburg-Landau Models Of Superconductivity With Normal Material Inclusion

In this paper, we study the relationship between the existence of the non-trivial solutions and the volume of normal materials for the Ginzburg-Landau models of superconductivity with normal material inclusion and obtain the following results:1) If the volume of normal materials is less than or equal to the volume of superconducting materials, the model always has non-trivial solutions when the applied magnetic field is small enough.2) If the volume of normal materials is greater than the volume of superconducting materials, there exists a critical value k~*. If the Ginzburg-Landau parameter k ≤ k~*, the model has only trivial solutions for any applied magnetic field; while, if k > k~*, there is another a critical value h~* = h~*(k) such that the model always has non-trivial solutions when the applied magnetic field is less than h~*.3) h~* → 0, as k →— k~* + 0.