Existence of multiple solutions to elliptic problems of Kirchhoff type with critical exponential growth (Q2877585)

This paper deals with elliptic problems of Kirchhoff type containing various nonlocal terms in \(\mathbb{R}^N\), \(N\geq 2\). Explicitly, in view of the Trudinger-Moser inequality, the author assumes that \(f(x,u)\) participating in the equation behaves like \(\exp(\alpha(x)|u|^{{N\over N-1}})\) for \(|u|\to\infty\), i.e. has a critical exponential growth. Using variational tools it is proved in Theorem 1.2 that the equation under consideration admits at least one nontrivial nonnegative weak solution in \(W^{1,N}(\mathbb{R}^N)\), while Theorem 1.3 asserts under more restrictive conditions that the same Kirchhoff type equation has at least 2 nontrivial and nonnegative weak solutions. More precisely, the proof of Theorem 1.3 is based on the mountain pass theorem.