Quasilinear elliptic problems with critical exponents and Hardy terms. (English) Zbl 1210.35021

Summary: Let \(\Omega\ni 0\) be an open bounded domain, \(\Omega\subset\mathbb R^N\) \((N>p^2)\). We are concerned with the multiplicity of positive solutions of
\[ -\Delta_pu- \mu\frac{|u|^{p-2}u}{|x|^p}= \lambda|u|^{p-2}u+ Q(x)|u|^{p^*-2}u, \quad u\in W_0^{1,p}(\Omega), \]
\[ -\Delta_pu= -\text{div}\big(|\nabla u|^{p-2}\nabla u\big), \quad 1<p<N, \qquad p^*=\frac{Np}{n-p}, \quad 0<\mu<\bigg( \frac{N-p}{p}\bigg)^p,\;\lambda>0, \]
and \(Q(x)\) is a nonnegative function on \(\overline{\Omega}\). By investigating the effect of the coefficient of the critical nonlinearity, we, by means of variational method, prove the existence of multiple positive solutions.


35B33 Critical exponents in context of PDEs
35J61 Semilinear elliptic equations
Full Text: DOI


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