Ground states of elliptic problems involving non homogeneous operators (Q2822153)

The article deals with the ground states of a \(C^{1}\)-functional \(\Phi(u):X\to {\mathbb R}\), i.e. solutions \(u\) to the problem \(\Phi'(u)=0\), \(\Phi(u)=\min\{\Phi(v):\;v\;\text{is a critical point of }\Phi\} \). Here \(X\) is a reflexive Banach space. The homogeneity of the main part of the functional \(\Phi\) is not required. First, under some conditions on \(\Phi\) the authors provide an abstract theorem about the existence of a ground state. It is also proven that in the case of an even \(\Phi\) satisfying the Palais-Smale condition on a manifold \(N=\{u\in X\setminus \{0\}: \Phi'(u)u=0\}\), the functional \(\Phi\) has infinitely many pairs of critical points. The results are applied to the study of three elliptic problems NEWLINE\[NEWLINE \begin{aligned} -\text{div\,}(a(|\nabla u|^{p})|\nabla u|^{p-2}\nabla u))=f(u),\;-M(\int_{\Omega}|\nabla u|^{2})\Delta u= f(u),\\ -\sum_{i=1}^{n}\partial_{x_{i}}(|\partial_{x_{i}}u|^{p_{i}-2}\partial_{x_{i}}u)=f(u),\;x\in \Omega\subset {\mathbb R}^{n},\;u|_{\partial \Omega}=0. \end{aligned}NEWLINE\]NEWLINE The domain \(\Omega\) here is bounded. It is proven that under certain growth conditions on \(f\) and other functions and parameters occurring in these equations the Dirichlet problem for these equations is solvable and sometimes these problems admit infinitely many solutions.