Construction of solutions for a critical problem with competing potentials via local Pohozaev identities

Abstract

In this paper, we consider the following critical equation: [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] are two nonnegative and bounded functions. Using a finite-dimensional reduction argument and local Pohozaev type of identities, we show that if [Formula: see text], [Formula: see text] has a stable critical point [Formula: see text] with [Formula: see text] and [Formula: see text], then the above equation has infinitely many positive solutions, where [Formula: see text] is the unique positive solution of [Formula: see text] with [Formula: see text]. Combining the results of [S. Peng, C. Wang and S. Wei, Constructing solutions for the prescribed scalar curvature problem via local Pohozaev identities, to appear in J. Differential Equations; S. Peng, C. Wang and S. Yan, Construction of solutions via local Pohozaev identities, J. Funct. Anal. 274 (2018) 2606–2633], it implies that the role of stable critical points of [Formula: see text] in constructing bump solutions is more important than that of [Formula: see text] and that [Formula: see text] can influence the sign of [Formula: see text], i.e. [Formula: see text] can be nonnegative, different from that in [S. Peng, C. Wang and S. Wei, Constructing solutions for the prescribed scalar curvature problem via local Pohozaev identities, to appear in J. Differential Equations]. The concentration points of the solutions locate near the stable critical points of [Formula: see text] which include the case of a saddle point.