Existence and multiple solutions for a critical quasilinear equation with singular potentials

  • Shaowei Chen
  • Zhi-Qiang WangEmail author


We study the following quasilinear elliptic equations
$$-\Delta_p u + V(x)|u|^{p-2}u = K( x)|u|^{q-2}u \,\,{\rm in}\,\, \mathbb{R}^N$$
where 1 < p < N and \({q = p(N - ps/b)/(N - p)}\) with constants b and s such that b < p, b ≠  0, \({ 0 < \frac{s}{b} < 1}\). This exponent q behaves like a critical exponent due to the presence of the potentials even though \({p < q < p^*= \frac{pN}{N-p}}\) the Sobolev critical exponent. The potential functions V and K are locally bounded functions and satisfy that there exist positive constants L, C 1, C 2, D 1 and D 2 such that \({C_1 \leq |x|^{b}V(x) \leq C_2}\) and \({D_1 \leq |x|^{s}K(x) \leq D_2}\) for \({|x| \geq L}\). We prove that below some energy threshold, the Palais–Smale condition holds for the functional corresponding to this equation. And we show that the finite energy solutions of this equation have exponential decay like \({e^{-\gamma|x|^{1-b/p}}}\) at infinity. If V has a critical frequency, i.e., V −1(0) has a non-empty interior, we prove that
$$-\Delta_p u + \lambda V(x)|u|^{p-2}u = K(x)|u|^{q-2}u\,\, {\rm in}\,\, \mathbb{R}^N$$
has more and more solutions as \({\lambda\rightarrow+\infty.}\)

Mathematics Subject Classification

35J20 35J60 


Quasilinear elliptic equation Critical exponent Decaying or coercive potentials Exponential decay 


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Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.School of Mathematical SciencesHuaqiao UniversityQuanzhouPeople’s Republic of China
  2. 2.Center for Applied MathematicsTianjin UniversityTianjinPeople’s Republic of China
  3. 3.Department of Mathematics and StatisticsUtah State UniversityLoganUSA

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