Infinitely many solutions for double phase problem with unbounded potential in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e22" altimg="si5.svg"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msup></mml:math>

Abstract

We consider a double phase problem in RN −div(|∇u|p−2∇u+a(x)|∇u|q−2∇u)+V(x)(|u|p−2u+a(x)|u|q−2u)=f(x,u)with an unbounded potential V and reaction term f, which does not satisfy the Ambrosetti–Rabinowitz condition. A new functional setting was provided for this problem. Using the Fountain and Dual Fountain Theorem with Cerami condition, we obtain some existence of infinitely many solutions. Our result extends some recent work in the literature.

Topics & Concepts