Existence of positive solutions for a p-Schrödinger–Kirchhoff integro-differential equation with critical growth

Abstract:

We consider the p-Schrödinger–Kirchhoff-type equation (Figure presented.) for v∈W^1,p(R^N), where σ~(s)=λf(s)+|s|^{p^∗^-²}s,b≥0,a,ε,λ>0,β=p²-Np+N and 1<p<N≤p+1<p^∗-2,p^∗=pN/(N-p). We assume that M and f verify conditions like those considered by Wang et al; in particular, M={x∈R^N/M(x)=M<inf>0</inf>}≠∅,M<inf>0</inf>=infM>0. Thanks to a study of the ground state of the limit problem associated to (P<inf>ε</inf>), we prove, by the method of Nehari manifold, the existence of a positive ground state of (P<inf>ε</inf>). By a Ljusternik–Schnirelmann scheme it’s shown, for ε small and λ big, that (P<inf>ε</inf>) has at least cat(M,M<inf>δ</inf>) positive solutions, where M<inf>δ</inf>={x∈R^N/dist(x,M)<δ},δ>0.

Año de publicación:

2024

Keywords:

• 35J60
• 45K05
• Integro-differential equation
• Ljusternik–Schnirelmann theory
• Method of Nehari manifold
• p-Schrödinger–Kirchhoff-type equation

Fuente:

scopus