A quasi-linear elliptic equation with critical growth on compact Riemannian manifold without boundary

Abstract

Let (M, g) be an N-dimensional compact Riemannian manifold without boundary. When m is a positive integer strictly smaller than N, we prove that

$$\sup_{\|u\|_{m,{N/m}}\leq 1}\int\limits_Me^{\alpha_{N,m}|u|^{N/(N-m)}}{\rm d}v_g< \infty,$$

where ||u||_m,N/m is the usual Sobolev norm of ${u\in W^{m,N/m}(M)}$, and α _N,m is the best constant in Adams’ original inequality (Ann Math 128:385–398, 1988). This is a modified version of Adams’ inequality on compact Riemannian manifold which has been proved by Fontana (Comment Math Helv 68:415–454, 1993). Using the above inequality in the case when m = 1, we establish sufficient conditions under which the quasi-linear equation

$$-\Delta_Nu+\tau|u|^{N-2}u=f(x,u)$$

has a nontrivial positive weak solution in W ^1,N(M), where ${-\Delta_N u{\,=\,}-{\rm div}(|\nabla u|^{N-2}\nabla u), \tau >0 }$, and f (x,u) behaves like ${e^{\gamma |u|^{N/(N-1)}}}$ as ${|u|\rightarrow \infty}$ for some γ > 0.