Existence and nonexistence results for quasilinear elliptic equations involving the \(p\)-Laplacian. (English) Zbl 1118.35010

This paper is devoted to a careful analysis of positive weak solutions to the quasilinear equation \(-\Delta_pu=\lambda h(x)|x|^{-p}|u|^{q-1}u+g(x)|u|^{p^*-1}u\) in \({\mathbb R}^N\), where \(N\geq 3\), \(1<p<N\), \(0<q\leq p-1\), \(\lambda >0\), and \(p^*=Np/(N-p)\) denotes the critical Sobolev exponent. The functions \(g\) and \(h\) are variable potentials. The main results of the paper establish existence and nonexistence properties and the proofs combine several variational techniques, such as concentration-compactness, Pokhozhaev type arguments, Palais-Smale sequences, as well as adequate comparison principles.


35J60 Nonlinear elliptic equations
47J30 Variational methods involving nonlinear operators
35B33 Critical exponents in context of PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)