Session S02 - Diverse Aspects of Elliptic PDEs and Related Problems

Wednesday, July 21, 18:30 ~ 19:00 UTC-3

Multiple solutions for the fractional nonlinear Schrödinger equation

Salomón Alarcón

Universidad Técnica Federico Santa María, Chile   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

We study the equation \begin{equation*}\label{ppp} \varepsilon^{2s}(-\Delta)^s u +V(x)u -f(u)=0\quad\mbox{in }\mathbb{R}^N, \tag{$P$} \end{equation*} where $s\in (0,1)$, $p\in \big(1,\frac{N+2s}{N-2s}\big)$, $N> 2s$, $f(u)=|u|^{p-1}u$, $V\in L^{\infty}(\mathbb{R}^N)$ is such that $\inf_{\mathbb{R}^N}V>0$ and $\varepsilon>0$ is small. Via a reduction method, we construct positive and sign-changing solutions concentrating at a saddle point of $V$ and sign-changing solutions of ($P$) concentrating at a local minimum point of $V$ as $\varepsilon\rightarrow0$. To guarantee the existence we cannot neglect the interaction between peaks. The proof of all our results is based in a max-min scheme which uses topological degree, extending some known results from the nonlinear case involving the Laplacian operator.

References

[1] T. D'Aprile and A. Pistoia, Existence, multiplicity and profile of sign-changing clustered solutions of a semiclassical nonlinear Schrödinger equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), 1423-1451.

[2] T. D'Aprile and D. Ruiz, Positive and sign-changing clusters around saddle points of the potential for nonlinear elliptic problems, Math. Z. 268 (2011), 605-634.

[3] J. Dávila, M. Del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differential Equations 256 (2014), 858-892.

[4] X. Kang, J. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations, Adv. Differential Equations 5 (2000), 899-928.

Joint work with Antonella Ritorto (Universiteit Utrecht, Germany) and Analía Silva (Universidad Nacional de San Luis, Argentina).

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