I am trying to show the existence of classical solution for the following problem using the mountain pass theorem :
$$ \left\{ \begin{array}{ccccccc} u^{''} + \lambda u + u³ = 0 (0<t<\pi)\\ u^{'}(0) = u^{'}(\pi) = 0 \\ \hspace{-0.7cm} \\ \end{array} \right. $$
for $\lambda <0$
An idea is aply the theorem for the functional
$$ \varphi (u) = \displaystyle\int_{0}^{\pi} \frac{1}{2}|u^{'} |² + \lambda u + \frac{u⁴}{4} \ dt$$
where u $ \in C^{2}[0,\pi]$ with $u^{'}(0) = u^{'}(\pi) = 0$. But i think this idea is not going to work because that in general the theorem is aplied for a functional defined in a convenient Sobolev space. Sorry if the question is simple, i am beginner on variational methods
Someone can give me some hints to prove the existence of classical solution for this problem?
thanks in advance
You should set up the problem in the space $W^{1,2}(]0,\pi[)$. Moreover the functional should be: $$\varphi(u):=\int_0^\pi \left(\frac12|u'|^2-\lambda u^2-\frac{u^4}{4}\right)dt$$ which verifies the hypotheses of the Mountain Pass Theorem. Notice that you need $\lambda>0$ to get rid of the space of constant functions.
In this way you can prove the existence of a (nontrivial) weak solution $u$ for the problem, i.e.
$$ u\in W^{1,2}(]0,\pi[),\qquad\int_0^\pi u'v'dt=\int_0^\pi\left(\lambda u+u^3\right)vd t\quad\forall v\in W^{1,2}(]0,\pi[). \tag{E}$$
Then you need a regularity argument to show that weak solutions actually are classical solutions of the equation and verify the boundary condition. This is not difficult, due to the fact that you are in dimension $1$. One ingredient is the fact: $$u'\in L^2(]0,\pi[)\Rightarrow u\in C([0,1])$$ which can be iterated to show that solutions are arbitrarily smooth. To get the boundary conditions $u'(0)=0$ notice that in (E) you can take $v=v_n$ such that $v'_n=n1_{]0,1/n[}$ and let $n\to\infty$ ($1_A$ denotes the indicator of the set $A$). The same goes for the other condition $u'(\pi)=0$.