Multibumpsolutions of nonlinear Schrödinger equations with steep potential welland indefinite potential
Multibumpsolutions of nonlinear Schrödinger equations with steep potential welland indefinite potential
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DOI：
10.3934/dcds.2013.33.7
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发表时间：
2012-09
影响因子：
1.1
通讯作者：
T. Bartsch;Z. Tang
中科院分区：
文献类型：
--
作者：
T. Bartsch;Z. Tang
本文研究了方程$-\Delta u+（\lambda a（x）+a_0（x））u=| u| ^{p-2}u$，$x\in{\mathbb R}^N$;这里$p>2$，$p 0 $对于$x\in{\mathbb R}^N$\bar{\Omega}$。与大多数关于这个问题的论文不同，我们允许$a_0\in L^\infty（{\mathbb R}^N）$改变符号。使用变分方法，我们证明了多凹凸解$u_\lambda$的存在性，该解以$\lambda\到\infty$的方式定位在规定的孤立开子集$\Omega_1，\dots，\Omega_k\subset\Omega $附近。运算符$L_0：=-\Delta+a_0$可能在$\Omega_j$中具有负特征值，$u_\lambda$的每个凸起可能是符号变化的。
We are concerned with the existence of single- and multi-bump solutions of the equation $-\Delta u+(\lambda a(x)+a_0(x))u=|u|^{p-2}u$, $x\in{\mathbb R}^N$; here $p>2$, and $p0$ for $x\in{\mathbb R}^N$\$\bar{\Omega}$. Unlike most other papers on this problem we allow that $a_0\in L^\infty({\mathbb R}^N)$ changes sign. Using variational methods we prove the existence of multibump solutions $u_\lambda$ which localize, as $\lambda\to\infty$, near prescribed isolated open subsets $\Omega_1,\dots,\Omega_k\subset\Omega$. The operator $L_0:=-\Delta+a_0$ may have negative eigenvalues in $\Omega_j$, each bump of $u_\lambda$ may be sign-changing.