Semiclassical limit for a quasilinear elliptic field equation: one-peak and multipeak solutions

Abstract

This paper deals with the existence of one-bump and multibump solutions for the following nonlinear field equation: $$-\Delta u+V(h x)u-\Delta_{p}u+ W'(u)=0$$ where $u:\mathbb R^{N}\rightarrow\mathbb R^{N+1},$ $N\geq 2,$ $p>N,$ $h>0,$ the potential $V$ is positive and $W$ is an appropriate singular function. Existence results are established provided that $h$ is sufficiently small, and we find solutions exhibiting a concentration behaviour in the semiclassical limit (i.e., as $h\rightarrow 0^{+}$) at any prescribed finite set of local minima, possibly degenerate, of the potential. Such solutions are obtained as local minima for the associated energy functional. No restriction on the global behaviour of $V$ is required except that it is bounded below away from zero. In the proofs of these results we use a variational approach, and the method relies on the study of the behaviour of sequences with bounded energy, in the spirit of the concentration-compactness principle.