Morse theory methods for a class of quasi-linear elliptic systems of higher order

Abstract

We develop the local Morse theory for a class of non-twice continuously differentiable functionals on Hilbert spaces, including a new generalization of the Gromoll–Meyer’s splitting theorem and a weaker Marino–Prodi perturbation type result. They are applicable to a wide range of multiple integrals with quasi-linear elliptic Euler equations and systems of higher order.

A Appendix: Comparing Hypothesis \({\mathfrak {F}}_{2,N,1,n}\) with controllable growth conditions

It is easily checked that Hypothesis ${\mathfrak {F}}_{2,N,1,n}$ for $n\ge 2$ may be equivalently formulated as Hypothesis ${\mathfrak {F}}_{2,N,1,n}$.   Let $z=(z_1,\ldots ,z_N)\in {\mathbb {R}}^{N}$, $p=\left( p^i_\alpha \right) \in {\mathbb {R}}^{N\times n}$, where $1\le i\le N$ and $\alpha \in {\mathbb {N}}_0^n$ with $|\alpha |=1$. Let ${\overline{\Omega }}\times {\mathbb {R}}^N\times {\mathbb {R}}^{N\times n}\ni (x, z,p)\mapsto F(x, z,p)\in {\mathbb {R}}$ be twice continuously differentiable in (z, p) for almost all x, measurable in x for all values of (z, p), and $F(\cdot ,z,p)\in L^1(\Omega )$ for $(z,p)=0$. Let $\kappa _n=2n/(n-2)$ for $n>2$, and $\kappa _n\in (2,\infty )$ for $n=2$. The derivatives of F fulfill the following properties:

(i) $F_{z_i}(\cdot , 0)\in L^{\kappa _n/(\kappa _n-1)}$ and $F_{p^i_\alpha }(\cdot , 0)\in L^{2}$ for $i=1,\ldots ,N$ and $|\alpha |=1$.

(ii) There exist positive constants ${\mathfrak {g}}_1$, ${\mathfrak {g}}_2$ and $s\in (0, \frac{\kappa _n-2}{\kappa _n})$, $r_\alpha \in (0,\frac{\kappa _n-2}{2\kappa _n})$ for each $\alpha \in {\mathbb {N}}_0^n$ with $|\alpha |=1$, such that for $i,j=1,\ldots ,N$, $|\alpha |=|\beta |=1$,

The controllable growth conditions (abbreviated to CGC below) [27, page 40] (that is, the so-called ‘common condition of Morrey’ or ‘the natural assumptions of Ladyzhenskaya and Ural’tseva’ [27, page 38,(I)]) may be, in our notation, expressed as:

CGC: ${\overline{\Omega }}\times {\mathbb {R}}^N\times {\mathbb {R}}^{N\times n}\ni (x, z,p)\mapsto F(x, z,p)\in {\mathbb {R}}$ is of class $C^2$, and there exist positive constants $\nu , \mu , \lambda , M_1, M_2$, such that with $|z|^2:=\sum ^N_{l=1}|z_l|^2$ and $|p|^2:=\sum _{|\alpha |=1}\sum ^N_{k=1}|p^k_\alpha |^2$,

Moreover, if $F=F(x,p)$ does not depend explicitly on z, the first three lines are replaced by

$$\begin{aligned}&\nu \left( 1+|p|^2\right) -\lambda \le F(x,p) \le \mu \left( 1+|p|^2\right) \quad \hbox {and}\\&|F_{p^i_\alpha }(x,p)|,\quad |F_{p^i_\alpha x_l}(x,p)|\le \mu \left( 1+|p|^2\right) ^{1/2}. \end{aligned}$$

From these it is not hard to see

Proposition 4.22

CGC implies Hypothesis ${\mathfrak {F}}_{2,N,1,n}$.