<p>In this paper, we study a Dirichlet problem involving a nonlinear and nonhomogeneous differential operator. The driven operator in the problem is in fact the sum of a negative <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$p(\cdot )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-Laplace operator and a negative <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$q(\cdot )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-Laplace operator weighted by a positive parameter. In the reaction of the problem, we have a Carathéodory function, which depends on the solution and its gradient. Moreover, such function is composed with a continuous map defined on the variable exponent Sobolev space <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$W_0^{1, q(\cdot )}(D)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>W</mi> <mn>0</mn> <mrow> <mn>1</mn> <mo>,</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, which we call intrinsic operator. Making use of tools from the theory of pseudomonotone operators, we here establish the existence of at least one solution for the problem under consideration. Further, we produce an upper bound for the solutions to the problem. Then, we prove the boundedness, closedness and compactness of the related solution set.</p>

On $(p(\cdot ), q(\cdot ))$-Equations With Convection Term and Intrinsic Operator

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2026年5月12日

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Xiaomei Lu,
Jinlan Pan,
Yehui Peng,
Francesca Vetro

摘要

In this paper, we study a Dirichlet problem involving a nonlinear and nonhomogeneous differential operator. The driven operator in the problem is in fact the sum of a negative $p(\cdot )$ $p (\cdot)$ -Laplace operator and a negative $q(\cdot )$ $q (\cdot)$ -Laplace operator weighted by a positive parameter. In the reaction of the problem, we have a Carathéodory function, which depends on the solution and its gradient. Moreover, such function is composed with a continuous map defined on the variable exponent Sobolev space $W_0^{1, q(\cdot )}(D)$ $W_{0}^{1, q (\cdot)} (D)$ , which we call intrinsic operator. Making use of tools from the theory of pseudomonotone operators, we here establish the existence of at least one solution for the problem under consideration. Further, we produce an upper bound for the solutions to the problem. Then, we prove the boundedness, closedness and compactness of the related solution set.

On \((p(\cdot ), q(\cdot ))\)-Equations With Convection Term and Intrinsic Operator

摘要