I want to write a thesis proposal about the singularly perturbed problem \begin{cases} -\varepsilon^2 A(\varepsilon^{-n} ||u||_{L^q}^q)\Delta u +V(x)u =H(x)u^p \\ 0<u\in H^1(\mathbb R^n) \end{cases} One side, I want to study the concentrating solutions. On the other side, for corresponding evolution equation, I want to study the orbital stability.
Although, about the concentrating solution, I have done something. But about orbital stability, I know nothing. Besides, I know little about PDE. My classmate tell me I should talk about the background in thesis proposal. He give me a frame.
Talk about the results of equation without global item. I think $V,H$ should be replaced by constants, then, we have a Schrodinger equation $$ -\Delta u + \alpha u=\beta u^p $$ what's results I should talk ? And which papers are them in ? Seemly, this is a old problem, rencent papers don't talk about it.
Talk about the results of Kirchhoff equation with global item.Mainly talk about the results of singularly perturbed and evolution equation. But I really don't what results is should be talking. In fact, I think recent a paper ,about the singular perturbation problem of Kirchhoff, contains many reference, but I don't what is suitable for my thesis. Besides, I know nothing about the evolution equation of Kirchhoff.
Talk about the known results of my equation. I think I should talk about the specail global item $A$. For example, G.F.Carrier have done $A(s)=1+s$ and $q=2$. And M. Chipot and J.-F. Rodrigues have done $A(s)=1+s$ and $q=1$, and so on.
According to this frame, I want to know what results or paper I should talk? Maybe, this is a lazy question, but I really don't know how to start. There are too many paper about Schrodinger equation and Kirchhoff equation, and I can't distinguish what is suitable for me.