Positive solutions for some quasilinear equations with critical and supercritical growth
The goal of this paper is to study the number of solutions of some quasilinear problems. More precisely, the authors prove the existence and multiplicity of positive solutions for the problem \[ -\text{div}(a(\varepsilon x)|\nabla u|^{p-2}\nabla u)+ u^{p-1}= f(u)+ u^{p^*-1}\text{ in }\mathbb{R}^d,\;u\in W^{1,p}(\mathbb{R}^d), \] where \(\varepsilon> 0\) is a small parameter, \(2\leq p< d\), \(p^*= {dp\over d-p}\), \(a\) is a positive potential and \(f\) is a superlinear function. The authors obtain the existence of a ground state solution and relate the number of positive solutions with the topology of the set where \(a\) attains its minimum. To this end the authors use minimax theorems and Ljusternik-Schnirelmann theory.
scientific article; zbMATH DE number 5142216
scientific article