Normalized solutions for p-Kirchhoff equation with general nonlinearities on bounded domain

Shupei Shen
Xiaoming He

Abstract. In this paper, we study the existence of the solutions for the following \(p\)-Kirchhoff equation on bounded domain \[\begin{cases}-\Bigg(a+b\displaystyle\int_{\Omega}|\nabla u|^pdx\Bigg)\Delta_pu+\lambda|u|^{p-2}u=h(u), &x \in \Omega,\\ u=0, &x\in\partial\Omega,\end{cases}\] with prescribed mass \[\int_{\Omega}|u|^pdx=m^p,\] where \(2\leq p\lt 3\), \(a\gt 0\), \(b\gt 0\) are positive constants, \(h(u)\) is a general nonlinearity with Sobolev subcritical growth, \(\Delta_p u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)\) is the \(p\)-Laplacian operator, \(\Omega\subset\mathbb{R}^3\) is a bounded domain, \(\lambda\in \mathbb{R}\) appears as a Lagrange multiplier. First, we prove that the equation admits a positive solution which is a local minimizer when \(h\) is \(L^p\)-subcritical or \(L^p\)-critical at infinity by Brezis-Nirenberg technique. Moreover, we get the multiplicity result by using the genus theory. Next, we prove that the equation has a local minimizer when \(h\) is \(L^p\)-supercritical. Besides, by using the Pohozaev analysis, we prove that in this case, if \(\Omega\) is a star-shaped domain with respect to the origin, the equation admits a second solution which is mountain pass type. To the best of our knowledge, this work seems to be the first contribution on the existence of normalized solutions for \(p\)-Kirchhoff equation on bounded domain.

Keywords: \(p\)-Laplacian Kirchhoff equation, normalized solutions, existence and multiplicity, bounded domain.

Mathematics Subject Classification: 35A15, 35B33, 35J20, 35J60.

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