Peak solutions without non-degeneracy conditions. (English) Zbl 1197.35121
In this paper, a very important progress is made by the author in replacing the non-degeneracy hypothesis one imposes when using Lyapunov-Schmidt procedure to prove existence of peak solutions. Typically peak solutions are shown to exit for problems of type:
\[ \varepsilon^2 \Delta u = g (x, u (x)) \quad\text{in } \Omega \]
with either Dirichlet or Neumann boundrey conditions.
He proves that solutions with single peak (multiple peaks) exist by considering a limit problem as the parameter goes to zero for certain rescaled equation and the non-degeneracy is asked for solutions of a problem posed on whole of \(\mathbb R^n\). In this paper, the non-degeneracy is replaced by appropriate degree and Conley index hypothesis. Since these are local the approach holds promise of extending to more general equations than discussed in the paper, where only the autonomous case is discussed.
\[ \varepsilon^2 \Delta u = g (x, u (x)) \quad\text{in } \Omega \]
with either Dirichlet or Neumann boundrey conditions.
He proves that solutions with single peak (multiple peaks) exist by considering a limit problem as the parameter goes to zero for certain rescaled equation and the non-degeneracy is asked for solutions of a problem posed on whole of \(\mathbb R^n\). In this paper, the non-degeneracy is replaced by appropriate degree and Conley index hypothesis. Since these are local the approach holds promise of extending to more general equations than discussed in the paper, where only the autonomous case is discussed.
Reviewer: P. N. Srikanth (Bangalore)
MSC:
| 35J70 | Degenerate elliptic equations |
| 35J25 | Boundary value problems for second-order elliptic equations |
Keywords:
removal of non-degeneracy; Conley index; small diffusion; peak solutions on bounded domains; removal of invertibility assumptionsReferences:
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