A multiplicity result for asymptotically linear Kirchhoff equations

By Chao Ji, Fei Fang, Binlin Zhang. 2017

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Abstract In this paper, we study the following Kirchhoff type equation: - ( 1 + b ∫ ℝ N | ∇ u | 2 d x ) Δ u + u = a ( x ) f ( u )   in ℝ N , u ∈ H 1 ( ℝ N ) , -\bigg{(}1+b\int_{\mathbb{R}^{N}}\lvert\nabla u|^{2}\,dx\biggr{)}\Delta u+u=a(% x)f(u)\quad\text{in }\mathbb{R}^{N},\qquad u\in H^{1}(\mathbb{R}^{N}), where N ≥ 3 {N\geq 3} , b > 0 {b>0} and f ⁢ ( s ) {f(s)} is asymptotically linear at infinity, that is, f ⁢ ( s ) ∼ O ⁢ ( s ) {f(s)\sim O(s)} as s → + ∞ {s\rightarrow+\infty} . By using variational methods, we obtain the existence of a mountain pass type solution and a ground state solution under appropriate assumptions on a ⁢ ( x ) {a(x)} .

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/a9e28e5ddde18c58f499926c9d42c6b6c453e420