On a generalized critical point theory on gauge spaces and applications to elliptic problems on \(\mathbb{R}^N\) (Q5946917)

The author develops some critical point theory and applies it to obtain weak solutions \(u\in H^1_{\text{loc}}(\mathbb R^N)\) of the problem NEWLINE\[NEWLINE-\Delta u+a(x)u=g(x,u).\tag{1}NEWLINE\]NEWLINE Let \(B_n=\{x\in\mathbb R^N:|x|\leq n\}\) and \(\phi_N(u):= {1\over 2} \int_{B_n}(|\nabla u|^2+a u^2)-\int_{B_n}G(x,u)\) where \(G(x,u)=\int^u_0 g(x,s)ds\). The author works with the multivalued map \(\Phi:E:=L^{2^*}_{\text{loc}}(\mathbb R^N)\to\mathbb R^{\mathbb N}\cup\{\infty\}\) defined by \(\Phi(u)=\prod_{n\in\mathbb N}[\phi_n(u),\infty)\) for \(u\in H^1_{\text{loc}}(\mathbb R^N)\), and \(\Phi(u)=\infty\) for \(u\in E\setminus H^1_{\text{loc}}(\mathbb R^N)\). For maps of this type the notions of slope, critical point, linking, and a version of the Palais-Smale condition are defined. NEWLINENEWLINENEWLINEThe author proves a deformation theorem and some abstract critical point theorems. In the application to problem (1) she can treat situations where one cannot expect solutions in \(H^1(\mathbb R^N)\).