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
[< i> Local mountain passes for semilinear elliptic problems
in unbounded domains< /i> , Calc. Var. Partial Differential Equations < b> 4< /b> (1996), 121–137],
[< i> Semi-classical states for nonlinear Schrödinger equations< /i> , J. Funct. Anal. < b> 149< /b>
(1997), 245–265]
and
[< i> Multi-peak bound states for nonlinear Schrödinger equations< /i> , Ann. Inst. H. Poincaré
Anal. Non Linéaire < b> 15< /b> (1998), 127–149], where "local
mountain passes" are found in certain nonlinear Schrödinger equations.