Three weak solutions for nonlinear problems of anisotropic double phase type

The aim of this paper is to establish the existence of at least three distinct weak solutions for a class of nonlinear elliptic problems with double phase structure under anisotropic Neumann-type boundary conditions. The problem is formulated in the framework of anisotropic Sobolev spaces and is governed by the following system: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <m:mfenced close="" open="{"> <m:mrow> <m:mtable class="cases"> <m:mtr> <m:mtd columnalign="left"> <m:mo>−</m:mo> <m:munderover accent="true" accentunder="false"> <m:mrow> <m:mo>∑</m:mo> </m:mrow> <m:mrow> <m:mi>j</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:munderover> <m:mfrac> <m:mrow> <m:mi>∂</m:mi> </m:mrow> <m:mrow> <m:mi>∂</m:mi> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>j</m:mi> </m:mrow> </m:msub> </m:mrow> </m:mfrac> <m:mfenced close="]" open="["> <m:mrow> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mi>j</m:mi> </m:mrow> </m:msub> <m:mfenced close=")" open="("> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mfrac> <m:mrow> <m:mi>∂</m:mi> <m:mi>φ</m:mi> </m:mrow> <m:mrow> <m:mi>∂</m:mi> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>j</m:mi> </m:mrow> </m:msub> </m:mrow> </m:mfrac> </m:mrow> </m:mfenced> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mi>j</m:mi> </m:mrow> </m:msub> <m:mfenced close=")" open="("> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mfrac> <m:mrow> <m:mi>∂</m:mi> <m:mi>φ</m:mi> </m:mrow> <m:mrow> <m:mi>∂</m:mi> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>j</m:mi> </m:mrow> </m:msub> </m:mrow> </m:mfrac> </m:mrow> </m:mfenced> </m:mrow> </m:mfenced> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="left"> <m:mspace width="2em"/> <m:mo>+</m:mo> <m:mi>α</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mfenced close=")" open="("> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>φ</m:mi> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>φ</m:mi> <m:mo>+</m:mo> <m:mo stretchy="false">|</m:mo> <m:mi>φ</m:mi> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>q</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo>