Infinitely many positive solutions for the nonlinear Shcrodinger equations in $R^N$

By Jun-cheng Wei. 2008

82 citations recorded by the source corpus.

We consider the following nonlinear problem in $\R^N$ $$\label{eq} - \Delta u +V(|y|)u=u^{p},\quad u>0 {in} \R^N, u \in H^1(\R^N) $$ where $V(r)$ is a positive function, $1<p<\frac{N+2}{N-2}$. We show that if $V(r)$ has the following expansion: There are constants $a>0$, $m>1$, $\theta>0$, and $V_0>0$, such that \[ V(r)= V_0+\frac a {r^m} +O\bigl(\frac1{r^{m+\theta}}\bigr),\quad \text{as $r\to +\infty$,} \] then \eqref{eq} has {\bf infinitely many non-radial positive} solutions, whose energy can be made arbitrarily large.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a6531cd88e29202e413fc50a6c81b59edb177674

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Manuel del Pino
2. Jun-cheng Wei
3. W. Yao
4. Weiwei Ao
5. Monica Musso
6. A. Malchiodi
7. Angela Pistoia
8. Shusen Yan

In Mathematical Physics

1. Manuel del Pino
2. Jun-cheng Wei
3. W. Yao
4. Weiwei Ao
5. Monica Musso
6. Kazunaga Tanaka
7. Jaeyoung Byeon
8. Zhi-Qiang Wang

Citations in the corpus

• Infinitely many positive solutions for nonlinear equations with non-symmetric potentials (2012, background): “As far as we know, without periodicity nor smallness of the parameters, the first result on the existence of infinitely many positive solutions was due to Wei-Yan [46].”
• Intermediate reduction method and infinitely many positive solutions of nonlinear Schrödinger equations with non-symmetric potentials (2015, methodology): “This idea comes directly from [43] and goes back at least as far as to [30].”