Electron. J. Diff. Equ., Vol. 2014 (2014), No. 252, pp. 1-12.

Nontrivial solutions for asymmetric problems on R^N

Ruichang Pei, Jihui Zhang

We consider the elliptic equation
 -\Delta u+V(x)u= f(x,u), \quad x\in \mathbb{R}^n, \quad u\in
 H^1(\mathbb{R}^N),\; N\geq 2,
where $V(x)\in C(\mathbb{R}^N)$ and $V(x)\geq V_0>0$ for all $x\in \mathbb{R}^N$. The nonlinear term $f$ exhibits an asymmetric growth at $+\infty$ and $-\infty$ in $\mathbb{R}^N\ (N\geq 2)$. Namely, it is linear at $-\infty$ and superlinear at $+\infty$. However, it need not satisfy the Ambrosetti-Rabinowitz condition on the positive semiaxis. Some existence results for nontrivial solution are established by using the minimax methods combined with the improved Moser-Trudinger inequality.

Submitted October 5, 2014. Published December 4, 2014.
Math Subject Classifications: 35J60, 35J20, 35B38.
Key Words: Schrodinger equation; asymmetric problems; one side resonance; subcritical exponential growth.

Show me the PDF file (253 KB), TEX file, and other files for this article.

Ruichang Pei
School of Mathematics and Statistics
Tianshui Normal University
Tianshui 741001, China
email: prc211@163.com
Jihui Zhang
School of Mathematics and Computer Sciences
Nanjing, Normal University
Nanjing 210097, China
email: zhangjihui@njnu.edu.cn

Return to the EJDE web page