A theorem about representation of Palais-Smale sequence and its applications. (English) Zbl 1138.35311
Summary: We prove a theorem about the representation of the Palais-Smale sequence to a semilinear elliptic equation, then use it to obtain some new existence results of nontrivial solutions for some semilinear elliptic equations
\[
-\Delta u + u = | u| ^ {p-2}u \quad\text{and}\quad -\varepsilon^ 2 \Delta u + u = | u| ^ {p-2}u
\]
in \(H_ 0^ 1(\Omega)\) on an unbounded domain \(\Omega\).
MSC:
| 35J20 | Variational methods for second-order elliptic equations |
| 35B25 | Singular perturbations in context of PDEs |
| 35J60 | Nonlinear elliptic equations |
| 58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |
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