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On the existence of nodal solutions for singular one-dimensional $\varphi$-Laplacian problem with asymptotic condition
Sign-changing multi-peak solutions for nonlinear Schrödinger equations with critical frequency
| 1. | Department of Mathematics, School of Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan |
$-\epsilon^2 \Delta u+V(x)u=f(u), \quad u\in H^1(\mathbb R^N),$ $\qquad\qquad\qquad$ (*)$_\epsilon $
where $V$ satisfies $ V(x)\geq 0 $, liminf$_{ |x|\to \infty }V(x)>0$. $f(u)\in C^1(\mathbb R,\mathbb R)$ satisfies Ambrosetti--Rabinowitz condition and some properties and $\frac{f(u)}{u}$ is nondecreasing. We consider the case inf$_{x\in \mathbb R^N}V(x)>0$ and critical frequency case, that is, inf$_{x\in \mathbb R^N}V(x)=0$. We study the existence of sign-changing 2-peak solutions of (*)$_\epsilon$ whose one peak is positive, another peak is negative and both peaks concentrate to a same local minimum point of $V(x)$ as $\epsilon \to 0$.
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