Multiple single-peaked solutions of a class of semilinear Neumann problems via the category of the domain boundary
For a smooth domain \(\Omega\) with compact boundary we investigate the problem \(-d^2\Delta u+u=f(u)\) with Neumann boundary conditions, where f has superlinear but subcritical growth. Provided that \(d>0\) is sufficiently small we show the existence of at least \(cat(\partial\Omega)\) positive solutions with single maximum points that lie on \(\partial\Omega\). We replace the standard variational setting used in the case of homogeneous f by considering the restriction of the free functional to a suitable submanifold of the Sobolev Space \(H^1(\Omega)\).
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