American Institute of Mathematical Sciences

October  2019, 39(10): 5825-5846. doi: 10.3934/dcds.2019256

Sign-changing solutions for some nonhomogeneous nonlocal critical elliptic problems

 Departamento de Matemática, Universidad Técnica Federico Santa María, Avenida España 1680, Valparaíso, Chile

Received  September 2018 Revised  May 2019 Published  July 2019

We construct multiple sign-changing solutions for the nonhomogeneous nonlocal equation
 $(-\Delta_{\Omega})^{s} u = \left| u\right| ^{\frac{4}{N-2s}}u +\varepsilon f(x)\quad \mbox{in }\Omega,$
under zero Dirichlet boundary conditions in a bounded domain
 $\Omega$
in
 $\mathbb{R}^{N}$
,
 $N>4s$
,
 $s\in (0,1]$
, with
 $f\in L^{\infty}(\Omega)$
,
 $f\geq 0$
and
 $f\neq0$
. Here,
 $\varepsilon>0$
is a small parameter, and
 $(-\Delta_{\Omega})^{s}$
represents a type of nonlocal operator sometimes called the spectral fractional Laplacian. We show that the number of sign-changing solutions goes to infinity as
 $\varepsilon\rightarrow 0$
when it is assumed that
 $\Omega$
and
 $f$
have certain smoothness and possess certain symmetries, and we are also able to establish accurately the contribution of the nonhomogeneous term in the found solutions. Our proof relies on the Lyapunov-Schmidt reduction method.
Citation: Salomón Alarcón, Jinggang Tan. Sign-changing solutions for some nonhomogeneous nonlocal critical elliptic problems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5825-5846. doi: 10.3934/dcds.2019256
References:

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