Summary
In this paper, we prove that solutions minimizing the nonlinear functional
among the Sobolev spaceH 10 (Ω) are unique when Ω is bounded convex domain in ℝ2. This uniqueness's results is equivalent to saying that solutions obtained from the Mountain Pass Lemma for the equation Δu+u p=0 are unique. We also prove that the level set of the unique solution is strictly convex.
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Lin, CS. Uniqueness of least energy solutions to a semilinear elliptic equation in ℝ2 . Manuscripta Math 84, 13–19 (1994). https://doi.org/10.1007/BF02567439
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DOI: https://doi.org/10.1007/BF02567439