Least-energy Solutions to a Non-autonomous Semilinear Problem with Small Diffusion Coefficient
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Least-energy solutions of a non-autonomous semilinear problem with a small diffusion coefficient are studied in this paper. We prove that the solutions will develop single peaks as the diffusion coefficient approaches 0. The location of the peaks is also considered in this paper. It turns out that the location of the peaks is determined by the non-autonomous term of the equation and the type of the boundary condition. Our results are based on fine estimates of the energies of the solutions and some non-existence results for semilinear equations on half spaces with Dirichlet boundary condition and some decay conditions at infinity.