Uniqueness of least energy solutions to a semilinear elliptic equation in ℝ2
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Summary
In this paper, we prove that solutions minimizing the nonlinear functional among the Sobolev spaceH 0 1 (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.
$$\frac{{\smallint \left| {\nabla \varphi } \right|^2 }}{{(\smallint \varphi ^{p + 1} )^{2/p + 1} }}$$
Key words
Semilinear elliptic equation solutions at the least energy uniqueness Pohozaev's identityPreview
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References
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