| Abstract: |
| We consider the condensation phenomena of a positive solution to
semilinear Neumann problems. By singular perturbation some positive solution tends to zero except for some points, and its energy concentrates around those points. That is said to be concentration phenomena. In precedent studies if the domain of a Neumann problem has a smooth boundary, then the mean curvature plays an important role in concentration phenomena. Especially if a solution has the least positive critical value, that is, a least-energy solution, then the solution concentrates at the point at which the mean curvature attains its maximum.
In our talk we assume that a domain of our problem has a piecewise smooth boundary. Especially we consider the case that the profile of the boundary is like a cone around a non-smooth point. Then we expect that the angle of the cone plays a similar role to the mean curvature. In fact, for a least-energy solution, we can prove that when the dimension of the domain is 2 or 3. Namely the least-energy solution concentrates at the peak having the least angle. Moreover we can investigate the profile of the least-energy solution around the peak. |
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