On domains and their indexes with applications to semilinear elliptic equations (Q2467074)

The author presents a deep and thorough analysis on the connection between the geometrical and topological structure of a domain \(\Omega\) in \(\mathbb{R}^N\), \(N\geq 1\) and the questions of existence, nonexistence and multiplicity of solutions of the semilinear elliptic equation \(-\Delta u+ u= |u|^{p-2}\) in \(\Omega\), \(u\in H^1_0(\Omega)\) with \(2< p< {2N\over N-2}\). Proceeding step by step with definitions, lemmas, remarks and examples, the author states and proves (sometimes by referring to results in the literature) 43 theorems, using (among other means) variational methods and Sobolev space theory. To convey some impression on the rich contents of the paper let the essential section headings be cited: 2. Large domains, \(y\)-symmetric domains and Esteban-Lions domains. 3. Palais-Smale theory. 4. Indexes of domains. 5. Palais-Smale conditions. 6. Non-ground state domains and higher energy solutions. 7. Ground state solutions and ground state domains. 8. A symmetric domain with two same indexes. 9. A \(y\)-symmetric domain with two different indexes.