Variational Inequalities with Multivalued Lower Order Terms and Convex Functionals in Orlicz-Sobolev Spaces

Ge Dong, Xiaochun Fang

Type: Article

Publication Date: 2015-01-01

Citations: 7

DOI: https://doi.org/10.1155/2015/321437

Abstract

We consider the existence of solutions of variational inequality form. Find<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mi>u</mml:mi><mml:mo>∈</mml:mo><mml:mi>D</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>J</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>:</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi mathvariant="script">A</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:mi>v</mml:mi><mml:mo>-</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo>+</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>F</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:mi>v</mml:mi><mml:mo>-</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo>+</mml:mo><mml:mi>J</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:mi>J</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>≥</mml:mo><mml:mn>0</mml:mn></mml:math>,<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mo>∀</mml:mo><mml:mi>v</mml:mi><mml:mo>∈</mml:mo><mml:msup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo></mml:math>whose principal part is having a growth not necessarily of polynomial type, where<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mrow><mml:mi mathvariant="script">A</mml:mi></mml:mrow></mml:math>is a second-order elliptic operator of Leray-Lions type,<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mrow><mml:mi>F</mml:mi></mml:mrow></mml:math>is a multivalued lower order term, and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mrow><mml:mi>J</mml:mi></mml:mrow></mml:math>is a convex functional. We use subsupersolution methods to study the existence and enclosure of solutions in Orlicz-Sobolev spaces.

Locations

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