Existence of solutions for a double-phase variable exponent equation without the Ambrosetti-Rabinowitz condition
Jingjing Liu, Patrizia Pucci
Abstract
Abstract The article deals with the existence of a pair of nontrivial nonnegative and nonpositive solutions for a nonlinear weighted quasilinear equation in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <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:math> {{\mathbb{R}}}^{N} , which involves a double-phase general variable exponent elliptic operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="bold">A</m:mi> </m:math> {\bf{A}} . More precisely, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="bold">A</m:mi> </m:math> {\bf{A}} has behaviors like <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mrow> <m:mo stretchy="false">∣</m:mo> <m:mrow> <m:mi>ξ</m:mi> </m:mrow> <m:mo stretchy="false">∣</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>q</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:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>ξ</m:mi> </m:math> {| \xi | }^{q\left(x)-2}\xi if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">∣</m:mo> <m:mrow> <m:mi>ξ</m:mi> </m:mrow> <m:mo stretchy="false">∣</m:mo> </m:mrow> </m:math> | \xi | is small and like <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mrow> <m:mo stretchy="false">∣</m:mo> <m:mrow> <m:mi>ξ</m:mi> </m:mrow> <m:mo stretchy="false">∣</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>p</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:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>ξ</m:mi> </m:math> {| \xi | }^{p\left(x)-2}\xi if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">∣</m:mo> <m:mrow> <m:mi>ξ</m:mi> </m:mrow> <m:mo stretchy="false">∣</m:mo> </m:mrow> </m:math> | \xi | is large. Existence is proved by the Cerami condition instead of the classical Palais-Smale condition, so that the nonlinear term <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> f\left(x,u) does not necessarily have to satisfy the Ambrosetti-Rabinowitz condition.