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Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents. (English) Zbl 0956.35056

There are obtained results about existence and multiplicity of solutions of the following Dirichlet problem: \[ -\Delta_p u=\sum^n_{i=1} \frac{\partial}{\partial x_i} \left(|\nabla u|^{p-2} u_{x_i}\right)= \lambda |u|^{r-2}u + \mu \frac{|u|^{q-2}u}{|x|^s} \quad\text{in} \Omega,\qquad u\big|_{\partial\Omega}=0. \] where \(\lambda, \mu\) are two positive parameters and \(\Omega\) is \(a\) smooth bounded domain in \(\mathbb{R}^n\) containing 0 in its interior. The parameters \(p,q,r,s,n\) satisfy the following natural conditions: \[ 1<p<n,\;p\leq q\leq p^\ast(s)\equiv \frac{n-s}{n-p}p,\quad p\leq r\leq p^\ast=p^\ast(0)=\frac{np}{n-p} \] which are necessary for an application of the variational approach.

MSC:

35J70 Degenerate elliptic equations
47J30 Variational methods involving nonlinear operators
58E30 Variational principles in infinite-dimensional spaces
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