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Multiple positive solutions for a quasilinear elliptic problem involving critical Sobolev-Hardy exponents and concave-convex nonlinearities. (English) Zbl 1219.35108

Summary: Let \(\Omega\subset\mathbb R^N\) \((N\geq 3)\) be a bounded smooth domain containing the origin. In this paper, by using variational methods, the multiplicity of positive solutions is obtained for a quasilinear elliptic problem
\[ -\Delta_pu- \mu\frac{|u|^{p-2}u}{|x|^p}= \frac{|u|^{p^*(t)-2}}{|x|^t} u+\lambda \frac{|u|^{q-2}}{|x|^s}u, \quad u\in W_0^{1,p}(\Omega), \]
with Dirichlet boundary condition, where \(\Delta_pu= \text{div}(|\nabla u|^{p-2}\nabla u)\), \(1<p<N\), \(0\leq \mu< \bar\mu= (\frac{N-p}{p})^p\), \(\lambda>0\), \(0\leq s,t<p\), \(1\leq q<p\) and \(p^*(t)= \frac{p(N-t)}{N-p}\) is the critical Sobolev-Hardy exponent.

MSC:

35J62 Quasilinear elliptic equations
35J20 Variational methods for second-order elliptic equations
35B09 Positive solutions to PDEs
35J25 Boundary value problems for second-order elliptic equations
35B33 Critical exponents in context of PDEs
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