Existence and concentration of local mountain passes for a nonlinear elliptic field equation in the semi-classical limit

Abstract

In this paper we are concerned with the problem of finding solutions for the following nonlinear field equation $$ -\Delta u + V(hx)u-\Delta_{p}u+ W'(u)=0, $$ where $u:\mathbb R^{N}\rightarrow \mathbb R^{N+1}$, $N\geq3$, $p> N$ and $h> 0$. We assume that the potential $V$ is positive and $W$ is an appropriate singular function. In particular we deal with the existence of solutions obtained as critical (not minimum) points for the associated energy functional when $h$ is small enough. Such solutions will eventually exhibit some notable behaviour as $h\rightarrow 0^{+}$. The proof of our results is variational and consists in the introduction of a modified (penalized) energy functional for which mountain pass solutions are studied and soon after are proved to solve our equation for $h$ sufficiently small. This idea is in the spirit of that used in M. Del Pino and P. Felmer [Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations 4 (1996), 121–137], [Semi-classical states for nonlinear Schrödinger equations, J. Funct. Anal. 149 (1997), 245–265] and [Multi-peak bound states for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998), 127–149], where "local mountain passes" are found in certain nonlinear Schrödinger equations.