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A non-radially symmetric solution to a class of elliptic equation with Kirchhoff term
Jianqing Chen,Xiuli Tang
Keywords:Equation with Kirchhoff term; Non-radially symmetric solution; Variant variational identity
      We consider the following equation with Kirchhoff term $$ -\left(a+b\int_{\mathbb{R}^3} {|\nabla u|^2} dx \right) \Delta u + u =|u|^{p-2}u , \quad u \in H^1 (\mathbb{R}^3), $$ where $a, b$ are positive constants and $2 < p < 6$. By deducing a variant variational identity and a constraint set, we are able to prove the existence of a non-radially symmetric solution $u(x_1, x_2, x_3)$ for the full range of $p\in (2,6)$. Moreover this solution $u(x_1, x_2, x_3)$ is radially symmetric with respect to $(x_1,x_2)$ and odd with respect to $x_3$.