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

Abstract

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, C1, C2, D1 and D2 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 \({\lambda\rightarrow+\infty.}\)