Critical groups computations on a class of Sobolev Banach spaces via Morse index
The aim of this study is to estimate the critical groups of some functionals associated to a class of quasilinear elliptic problems, involving \(p\)-Laplacian. So, a class of Banach spaces which are the natural variational setting for several quasilinear elliptic problems is investigated. Precisely, the authors work on the Sobolev space \(W^{1,p}_0(\Omega)\) where \(2< p<\infty\), and \(\Omega\) is a bounded domain of \(\mathbb{R}^N\) \((N\geq 1)\), with sufficiently regular boundary \(\partial\Omega\). The functional \(f: W^{1,p}_0(\Omega)\to \mathbb{R}\) defined by setting \[ f(u)= {1\over p} \int_\Omega|\nabla u|^p dx+{1\over 2} \int_\Omega|\nabla u|^2 dx+\int_\Omega G(u) dx, \] where \(G(t)= \int^t_0 g(s) ds\) and \(g\in C^1(\mathbb{R}, \mathbb{R})\) satisfies the following assumption: \(|g'(t)|\leq c_1|t|^q+ c_2\) with \(c_1,c_2\) positive constants and \(0\leq q< p^*- 2\) if \(N> p\), while \(q\) is any positive number if \(N= p\). Otherwise, if \(N< p\), no restrictive assumption on the growth of \(g\) is required. The critical points of \(f\) in \(W^{1,p}_0(\Omega)\) correspond to the weak solution of the quasilinear elliptic problem \[ \begin{cases} -\Delta_p u-\Delta u+ g(u)= 0\quad &\text{in }\Omega,\\ u= 0\quad &\text{on }\partial\Omega,\end{cases} \] where \(\Delta_p u=\text{div}(|\nabla u|^{p-2}\nabla u)\), \(p> 2\), arising in the mathematical desription of propagation phenomena of solitary waves. In spite of the difficulties related to the fact that any critical point \(u\) of \(f\) is degenerate and \(f''(u)\) is not a Fredholm operator, so that the generalized splitting Morse lemma cannot be applied in order to describe the behavior of \(f\) near the critical points, the authors obtain critical groups estimates for functionals \(f\) in \(u\), in the spirit of the generalized Morse lemma.
scientific article; zbMATH DE number 1912451
scientific article