Nonlinear scalar field equations with Berestycki-Lions’ nonlinearity on large domains. (English) Zbl 1454.35153
Summary: We prove the existence of solutions for the following semilinear elliptic equation:
\[ - \Delta u = g(u) \quad \text{in } \Omega, \quad u \in H^1_0(\Omega). \]
Here \(\Omega\) is a suitable large domain and \(g\) satisfies the completely same conditions as Berestycki-Lions’ conditions. Those conditions of \(g\) are known as “almost sufficient and necessary conditions” to the existence of nontrivial solutions of the equations defined in \(\mathbb{R}^N\). The main difficulty to prove the existence of solutions of the equation is that we can not obtain bounded Palais-Smale sequences. To overcome this difficulty, we modify the corresponding functional, which is a new idea introduced in our previous paper.
MSC:
| 35J61 | Semilinear elliptic equations |
| 35J20 | Variational methods for second-order elliptic equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
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