Symmetry and nonexistence of low Morse index solutions in unbounded domains. (English) Zbl 1189.35100
Summary: We prove symmetry results for classical solutions of semilinear elliptic equations in the whole \(\mathbb R^N\) or in the exterior of a ball, \(N \geqslant 2\), in the case when the nonlinearity is either convex or has a convex first derivative. More precisely, we prove that solutions having Morse index \(j\leqslant N\) are foliated Schwarz symmetric, i.e., they are axially symmetric with respect to an axis passing through the origin and nonincreasing in the polar angle from this axis. From this, we deduce some nonexistence results for positive or sign changing solutions in the case when the nonlinearity does not depend explicitly on the space variable.
MSC:
| 35J61 | Semilinear elliptic equations |
| 35J91 | Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian |
| 35B07 | Axially symmetric solutions to PDEs |
Keywords:
semilinear elliptic equations; symmetry results; unbounded domains; nonexistence of solutionsReferences:
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