Spikes of sign-changing solutions to the critical Schrödinger equations with trapping potentials

ABSTRACT

Consider the following Schrödinger equation: $$\begin{array}{c}\hfill \left\{\begin{array}{c}-{\mathit{\epsilon }}^2\mathrm{\Delta }u+V(x)u={\mathit{\alpha }|u|}^{p-2}u+{|u|}^{2^{\ast }-2}u\text{in}{\mathbb{R}}^N,\hfill \\ u\in H^1({\mathbb{R}}^N),\hfill \end{array}\right.\end{array}$$

where $N\ge 4$, V(x) is a trapping potential, $\mathit{\alpha }>0$ is a constant, $\mathit{\epsilon }>0$ is a small parameter and $2<p<2^{\ast }=\frac{2N}{N-2}$. By using the variational method, we obtain a least energy sign-changing solution to this equation for $\mathit{\epsilon }>0$ small enough. The concentration behavior of this sign-changing solution as $\mathit{\epsilon }\to 0^{+}$ is also studied. Furthermore, by combining the uniformly elliptic estimates and local energy estimates, we also obtain the location of the spikes as $\mathit{\epsilon }\to 0^{+}$. To the best of our knowledge, this is the first attempt devoted to the spikes of sign-changing solutions to Schrödinger equations with Sobolev critical exponent and trapping potentials.

KEYWORDS:

• Sign-changing solution
• Sobolev critical exponent
• spike
• trapping potential
• variational method

AMS SUBJECT CLASSIFICATIONS:

• 35B09
• 35B33
• 35J50

Acknowledgements

Y. Wu also would like to thank the anonymous referees for carefully reading the manuscript and valuable comments that greatly helped improve this paper.

Notes

No potential conflict of interest was reported by the author.

Additional information

Funding

Y. Wu was supported by the Fundamental Research Funds for the Central Universities [grant number 2017XKQY091].