Ground state solution for the Choquard equation under the superposition of operators of mixed fractional order

Abstract We establish the existence of a ground state solution for the fractional Choquard equation governed by the superposition operator $$ \int _{[0, 1]} (-\varDelta )^s u\, d\mu (s), $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mo>∫</mml:mo> <mml:mrow> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> </mml:msub> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>s</mml:mi> </mml:msup> <mml:mi>u</mml:mi> <mml:mspace/> <mml:mi>d</mml:mi> <mml:mi>μ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> and in presence of a confining potential. Here, $$\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>μ</mml:mi> </mml:math> denotes a signed measure on the interval of fractional exponents [0, 1]. The presence of the superposition operator asks for the problem to be addressed with a special care. However, the wide generality of this setting allows to provide entirely new existence results in several special cases of interest, including, e.g., the mixed operator one $$-\varDelta + (-\varDelta )^s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>+</mml:mo> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>s</mml:mi> </mml:msup> </mml:mrow> </mml:math> . We point out that the possibility of considering operators “with the wrong sign" is also a complete novelty.