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On the quasilinear elliptic problems with critical Sobolev-Hardy exponents and Hardy terms. (English) Zbl 1135.35042

Summary: Let \(\Omega\subset\mathbb R^N\) be a smooth bounded domain such that \(0\in\Omega\), \(N\geq 3\). In this paper, we study the critical quasilinear elliptic problems
\[ -\Delta_pu-\mu \frac{|u|^{p-2}u}{|x|^p}= \frac{|u|^{p^*(t)-2}}{|x|^t}u+\lambda \frac{|u|^{q-2}}{|x|^s}u, \qquad u\in W_0^{1,p}(\Omega) \]
with Dirichlet boundary condition, where \(-\Delta_pu= -\text{div}(|\nabla u|^{p-2} \nabla u)\), \(1<p<N\), \(0\leq\mu<\overline{\mu}:= (\frac{N-p}{p})^p\), \(\lambda>0\), \(0\leq s\), \(t<p\), \(p\leq q< p^*(s):= \frac{p(N_s)}{N-p}\), \(p^*(t):= \frac{p(N-t)}{N-p}\), \(p^*(s)\) and \(p^*(t)\) are the critical Sobolev-Hardy exponents. Via variational methods, we deal with the conditions that ensure the existence of positive solutions for the equation. The results depend crucially on the parameters \(p\), \(q\), \(s\), \(\lambda\) and \(\mu\).

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35J60 Nonlinear elliptic equations
35B33 Critical exponents in context of PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)
35J35 Variational methods for higher-order elliptic equations
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References:

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