On a Kirchhoff type problems with potential well and indefinite potential

2015/07/13 by Yuanze Wu, Yisheng Huang, Wu, Yuanze +3
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.1507.03373

13

arxiv created 2015/07/13 · arxiv updated 2015/07/14

Abstract

In this paper, we study the following Kirchhoff type problem:% \\aligned-(α∫_\bbr³|∇ u|²dx+1)Δu+(λa(x)+a₀)u=|u|^p-2u in \bbr³,
% u∈\h,\endaligned.\eqno(P_α,λ)% where 4<p<6, α and λ are two positive parameters, a₀∈\bbr is a (possibly negative) constant and a(x)≥0 is the potential well. By the variational method, we investigate the existence of nontrivial solutions to (P_α,λ). To our best knowledge, it is the first time that the nontrivial solution of the Kirchhoff type problem is found in the indefinite case. We also obtain the concentration behaviors of the solutions as λ→+∞.

Related

• On finding solutions of a Kirchhoff type problem
• Existence of solutions for a class of Kirchhoff-type equations with indefinite potential
• Normalized Solutions to Kirchhoff Equation with Nonnegative Potential
• Multi-bump solutions for a Kirchhoff problem type
• On a critical Kirchhoff problem in high dimensions