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

Nontrivial solutions for asymmetric problems on R^N

Ruichang Pei, Jihui Zhang

Abstract:
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.

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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

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