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Necessary and sufficient conditions for ground state solutions to planar Kirchhoff-type equations
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View
- Published online by Cambridge University Press:
- 11 March 2024, pp. 1-22
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In this paper, we are concerned with the ground states of the following planar Kirchhoff-type problem:
\[ -\left(1+b\displaystyle\int_{\mathbb{R}^2}|\nabla u|^2\,{\rm d}x\right)\Delta u+\omega u=|u|^{p-2}u, \quad x\in\mathbb{R}^2. \]where $b,\, \omega >0$
are constants, $p>2$
. Based on variational methods, regularity theory and Schwarz symmetrization, the equivalence of ground state solutions for the above problem with the minimizers for some minimization problems is obtained. In particular, a new scale technique, together with Lagrange multipliers, is delicately employed to overcome some intrinsic difficulties.
