2015/07/11 by Huang, Yisheng, Liu, Zeng, Wu, Yuanze
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1507.03056
In this paper, we study the following biharmonic equations:% \\alignedamp;Δ2u-a0Δu+(λb(x)+b0)u=f(u)amp; in \bbrN,
% amp;u∈\h,\endaligned.\eqno(Pλ)% where N≥3, a0,b0∈\bbr are two constants, λ>0 is a parameter, b(x)≥0 is a potential well and f(t)∈ C(\bbr) is subcritical and superlinear or asymptotically linear at infinity. By the Gagliardo-Nirenberg inequality, we make some observations on the operator Δ2-a0Δ+λb(x)+b0 in \h. Based on these observations, we give a new variational setting to (Pλ) for a0<0. With this new variational setting in hands, we establish some new existence results of the nontrivial solutions to (Pλ) for all a0, b0∈\bbr with λ sufficiently large by the variational method. The concentration behavior of the nontrivial solutions as λ→+∞ is also obtained. It is worth to point out that it seems to be the first time that the nontrivial solution of (Pλ) is obtained in the case of a0<0.