×

Infinitely many small solutions for the \(p\)& \(q\)-Laplacian problem with critical Sobolev and Hardy exponents. (English) Zbl 1278.35116

Summary: We study the following \(p\)& \(q\)-Laplacian problem with critical Sobolev and Hardy exponents \[ \begin{cases} -\Delta_p u - \Delta_q u = \mu \frac{|u|^{p^\ast(s)-2}u}{|x|^s} + \lambda f(x, u) & \text{ in } \Omega,\\ u = 0 & \text{ on } \Omega,\end{cases} \]
where \(\Omega \subset \mathbb{R}^N\) is a bounded domain and \(\Delta_r u = \mathrm{div}(|\nabla u|^{r-2}\nabla u)\) is the \(r\)-Laplacian of \(u\). By using the variational method and concentration-compactness principle, we obtain the existence of infinitely many small solutions for above problem which are the complement of previously known results.

MSC:

35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35B33 Critical exponents in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
PDFBibTeX XMLCite