### Sign changing solutions of nonlinear Schrödinger equations

DOI: http://dx.doi.org/10.12775/TMNA.1999.010

#### Abstract

We are interested in solutions $u\in H^1({\mathbb R}^N)$ of the linear Schrödinger equation

$-\delta u +b_{\lambda} (x) u =f(x,u)$. The nonlinearity $f$ grows

superlinearly and subcritically as $\vert u\vert \to\infty$.

The potential $b_{\lambda}$ is positive, bounded away from $0$, and has a potential well.

The parameter $\lambda$ controls the steepness of the well.

In an earlier paper we found a positive and a negative solution.

In this paper we find third solution. We also prove that this third solution

changes sign and that it is concentrated in the potential well if $\lambda \to \infty$.

No symmetry conditions are assumed.

$-\delta u +b_{\lambda} (x) u =f(x,u)$. The nonlinearity $f$ grows

superlinearly and subcritically as $\vert u\vert \to\infty$.

The potential $b_{\lambda}$ is positive, bounded away from $0$, and has a potential well.

The parameter $\lambda$ controls the steepness of the well.

In an earlier paper we found a positive and a negative solution.

In this paper we find third solution. We also prove that this third solution

changes sign and that it is concentrated in the potential well if $\lambda \to \infty$.

No symmetry conditions are assumed.

#### Keywords

Nonlinear Schrödinger equation; sign changing solution; localized solution; potential well; singular limit

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