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Multiplicity of solutions for a quasilinear problem with supercritical growth. (English) Zbl 1166.35301

Summary: The multiplicity and concentration of positive solutions are established for the equation \[ -\varepsilon^{p}\Delta_{p}u+V(z)|u|^{p-2}u=|u|^{q-2}u +\lambda|u|^{s-2}u \quad\text{in }\mathbb{R}^N, \] where \(1<p<N\), \(\varepsilon >0\), \(p<q<p^* \leq s\), \(p^{*}=\frac{Np}{N-p}\), \(\lambda\geq 0\) and \(V\) is a positive continuous function.

MSC:

35A15 Variational methods applied to PDEs
35H30 Quasielliptic equations
35B40 Asymptotic behavior of solutions to PDEs
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