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arXiv · 1611.07632

Singularly perturbed fractional Schrödinger equation involving a general critical nonlinearity

Hua Jin·Wenbin Liu·Jianjun Zhang
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Abstract

In this paper, we are concerned with the existence and concentration phenomena of solutions for the following singularly perturbed fractional Schrödinger problem \begin{align*} \varepsilon^{2s}(-Δ)^su+V(x)u=f(u) \ \ \ \mbox{in} \ \ \ \mathbb{R}^N, \end{align*} where $N>2s$ and the nonlinearity $f$ has critical growth. By using the variational approach, we construct a localized bound-state solution concentrating around an isolated component of the positive minimum point of $V$ as $\varepsilon\rightarrow 0$. Our result improves the study made in X. He and W. Zou ({\it Calc. Var. Partial Differential Equations}. 55-91(2016)), in the sense that, in the present paper, the {\it Ambrosetti-Rabinowitz} condition and {\it monotonicity} condition on $f(t)/t$ are not required.

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Hua Jin, Wenbin Liu, Jianjun Zhang. 2017-02-08. Singularly perturbed fractional Schrödinger equation involving a general critical nonlinearity. https://arxiv.org/abs/1611.07632

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