Multiplicity of solutions for a Kirchhoff equation with subcritical or critical growth.
In this article, the existence of infinitely many weak solutions for the Dirichlet problem involving nonlocal equations of Kirchhoff type is investigated. In particular, the authors consider the following problem: \[ - M\left (\int _{\Omega }| \nabla u| ^2 \text{d}x \right ) \Delta u = \lambda | u| ^{q-2}u +| u| ^{p-2}u \, \text{ in } \Omega ,\; u=0 \text{ on } \partial \Omega , \] where \(\Omega \) is a smooth domain in \(\mathbb {R}^N\), with \(N\in \{1,2,3\}\), \(M\:\mathbb {R}_{+}\rightarrow \mathbb {R}_{+}\) is a continuous function, \(\lambda \) is a positive parameter, and the exponents \(q,p\) are such that \(1<q<2<p\), with \(p\leq 6\) if \(N=3\). For \(N=3\), the subcritical cases \(2<p<4\), \(p=4\), \(4<p<6\) and the critical case \(p=6\) are singly examinated and related existence results of infinitely many weak solutions are obtained under suitable conditions on the function \(M\). The existence of infinitely many weak solutions is established for all \(\lambda >0\) when \(2<p\leq 4\), and for all \(\lambda >0\) small enough when \(4<p\leq 6\). The proofs follow standard arguments based on variational methods. The key tool is a consequence of a well known abstract result by \textit{D. C. Clark} [Math. J., Indiana Univ. 22, 65--74 (1972; Zbl 0228.58006)] on the existence of infinitely critical points for \(C^1\)-functionals on Banach spaces which satisfy the Palais-Smale condition and are invariant under the action of a compact topological group.
scientific article; zbMATH DE number 6152276