In the present work we are interested in studying the existence and multiplicity of solutions for the following class of quasilinear problems where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ u: \mathbb {R}^N \rightarrow \mathbb {R}^{N+1} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo>:</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> <mo stretchy="false">→</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ N \ge 3 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Phi :\mathbb {R}\rightarrow [0,+\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Φ</mi> <mo>:</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">→</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mo>+</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is a N-function, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ \Delta _\Phi $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Δ</mi> <mi mathvariant="normal">Φ</mi> </msub> </math></EquationSource> </InlineEquation> is the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ \Phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation>-Laplacian operator and b, V and W are functions satisfying some technical conditions.

Existence and multiplicity of solutions for a class of quasilinear elliptic field in Orlicz-Sobolev spaces

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2025年11月4日

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Claudianor O. Alves,
Cícero J. da Silva

摘要

In the present work we are interested in studying the existence and multiplicity of solutions for the following class of quasilinear problems where $ u: \mathbb {R}^N \rightarrow \mathbb {R}^{N+1} $ $u : R^{N} \to R^{N + 1}$ , $ N \ge 3 $ $N \geq 3$ , $\Phi :\mathbb {R}\rightarrow [0,+\infty )$ $Φ : R \to [0, + \infty)$ is a N-function, $ \Delta _\Phi $ $Δ_{Φ}$ is the $ \Phi $ $Φ$ -Laplacian operator and b, V and W are functions satisfying some technical conditions.