| Summary: | We are concerned with the existence and multiplicity of normalized solutions to the fractional Schrödinger equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="{" close=""><mtable><mtr><mtd columnalign="left"><mrow><msup><mrow><mo>(</mo><mo>−</mo><mo>Δ</mo><mo>)</mo></mrow><mi>s</mi></msup><mi>u</mi><mo>+</mo><mi>V</mi><mrow><mo>(</mo><mi>ε</mi><mi>x</mi><mo>)</mo></mrow><mi>u</mi><mo>=</mo><mi>λ</mi><mi>u</mi><mo>+</mo><mi>h</mi><mrow><mo>(</mo><mi>ε</mi><mi>x</mi><mo>)</mo></mrow><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mrow><mrow><mi>i</mi><mi>n</mi></mrow><mspace width="4pt"></mspace><msup><mi mathvariant="double-struck">R</mi><mi>N</mi></msup><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mo>∫</mo><msup><mi mathvariant="double-struck">R</mi><mi>N</mi></msup></msub><msup><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow><mn>2</mn></msup><mi>d</mi><mi>x</mi><mo>=</mo><mi>a</mi><mo>,</mo></mrow></mstyle></mtd></mtr></mtable></mfenced></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow><mo>(</mo><mo>−</mo><mo>Δ</mo><mo>)</mo></mrow><mi>s</mi></msup></semantics></math></inline-formula> is the fractional Laplacian, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>,</mo><mi>ε</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula> is an unknown parameter that appears as a Lagrange multiplier, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>h</mi><mo>:</mo><msup><mi mathvariant="double-struck">R</mi><mi>N</mi></msup><mo>→</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula> are bounded and continuous, and <i>f</i> is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>2</mn></msup></semantics></math></inline-formula>-subcritical. Under some assumptions on the potential <i>V</i>, we show the existence of normalized solutions depends on the global maximum points of <i>h</i> when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ε</mi></semantics></math></inline-formula> is small enough.
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