Normalized solutions to nonlinear Schrödinger equations with mixed fractional Laplacians

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> \{{array}{l}{(-Δ )}^{{s}₁}u+{(-Δ )}^{{s}₂}u+λ u=f(u){1.0em}{in}{0.33em}{{R}}N,{1.0em}\\ { ∫ }_{{{R}}N}{| u(x)| }²{d}x=c,{1.0em}\\ u∈ {H}^{{s}₂}({{R}}N),{1.0em}{array}. where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>0</m:mn> <m:mo><</m:mo> <m:msub> <m:mrow> <m:mi>s</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo><</m:mo> <m:msub> <m:mrow> <m:mi>s</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mo><</m:mo> <m:mn>1</m:mn> </m:math> 0< {s}₁< {s}₂< 1 , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <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:mi>i</m:mi> </m:mrow> </m:msub> </m:mrow> </m:msup> </m:math> {(-Δ )}^{{s}ᵢ} is fractional Laplacian. We get a sharp description of the existence and nonexistence of the global minimizer on the mass constraint, which is called energy ground state. By using of concentration-compactness principle, we show that there exists a constant <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> </m:math> {c}₀ such that there exists an energy ground state if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>c</m:mi> <m:mo>></m:mo> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> </m:math> c> {c}₀ and there exists no energy ground state if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>0</m:mn> <m:mo><</m:mo> <m:mi>c</m:mi> <m:mo><</m:mo> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> </m:math> 0< c< {c}₀ . Some almost critical conditions which determine <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:math> {c}₀=0 or <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:mo>></m:mo> <m:mn>0</m:mn> </m:math> {c}₀> 0 are established. In the case <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>c</m:mi> <m:mo>=</m:mo> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> </m:math> c={c}₀ , under certain conditions, we establish the existence result of energy ground state. Finally, by using the method of mountain pass type characterization, we show that any energy ground state is action ground state of the corresponding action functional.