Normalized solutions to nonlinear Schrödinger equations with mixed fractional Laplacians

Abstract We consider the following elliptic problem with mixed local and nonlocal operators: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mfenced open="{" close=""> <m:mrow> <m:mtable displaystyle="true"> <m:mtr> <m:mtd columnalign="left"> <m:msup> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mo>−</m:mo> <m:mi>Δ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>s</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msup> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:msup> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mo>−</m:mo> <m:mi>Δ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>s</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msup> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mi>λ</m:mi> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mspace width="1.0em"/> <m:mi mathvariant="normal">in</m:mi> <m:mspace width="0.33em"/> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> <m:mo>,</m:mo> <m:mspace width="1.0em"/> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="left"> <m:munder> <m:mrow> <m:mrow> <m:mstyle displaystyle="true"> <m:mo>∫</m:mo> </m:mstyle> </m:mrow> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> </m:mrow> </m:munder> <m:msup> <m:mrow> <m:mo>∣</m:mo> <m:mi>u</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi mathvariant="normal">d</m:mi> <m:mi>x</m:mi> <m:mo>=</m:mo> <m:mi>c</m:mi> <m:mo>,</m:mo> <m:mspace width="1.0em"/> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="left"> <m:mi>u</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mi>H</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>s</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mspace width="1.0em"/> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:math> \left\{\begin{array}{l}{\left(-\Delta )}^{{s}_{1}}u+{\left(-\Delta )}^{{s}_{2}}u+\lambda u=f\left(u)\hspace{1.0em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{N},\hspace{1.0em}\\ \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{| u\left(x)| }^{2}{\rm{d}}x=c,\hspace{1.0em}\\ u\in {H}^{{s}_{2}}\left({{\mathbb{R}}}^{N}),\hspace{1.0em}\end{array}\right. where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>0</m:mn> <m:mo>&lt;</m:mo>