Eigenvalue problems for quasilinear elliptic systems with limiting nonlinearity

Abstract

In this paper, we consider a class of quasilinear elliptic eigenvalue problems with limiting nonlinearity. First, we use the concentration-compactness principle to get the existence of a minimum uεH ¹₀ (ω,R ^N) of the minimization problem$I_{\lambda _0 } = \inf \{ \smallint _\Omega (a_{\alpha \beta } (x)g_{ij} (u)D_\alpha u^i D_\beta u^j + h(x)|u|^2 )|u \in H_0^1 (\Omega ,R^N ),\smallint _\Omega |u|^{2n/(n - 2)} = \lambda _0 \} ;$ then we apply the reverse Hölder inequality to prove thatuεL ^∞ (ω, R ^N).