This paper investigates the existence of energy solutions for a Schrödinger-Maxwell system with a variable exponent on a bounded domain <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {D}\subset \mathbb {R}^N$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">D</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$N\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>), where the p(x)-Laplacian couples the unknown functions u and v. Under the assumptions that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$f\in L^{m(\cdot )}(\mathcal {D})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <msup> <mi>L</mi> <mrow> <mi>m</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">D</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$m(x)>N/p(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>></mo> <mi>N</mi> <mo stretchy="false">/</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and that the continuous exponent satisfies <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$p(\cdot )>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, a variational approach is employed to establish the existence of weak solutions <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$(u,v)\in \left( W_{0}^{1,p(\cdot )}(\mathcal {D})\right) ^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msup> <mfenced close=")" open="("> <msubsup> <mi>W</mi> <mrow> <mn>0</mn> </mrow> <mrow> <mn>1</mn> <mo>,</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">D</mi> <mo stretchy="false">)</mo> </mrow> </mfenced> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> for the considered system. Furthermore, it is shown that this solution is a saddle point of the associated energy functional.

Existence of weak solutions for variable exponent Schrödinger–Maxwell systems

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发表日期：

2026年7月11日

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Hichem Khelifi,
Mohamed Amine Zouatini

摘要

This paper investigates the existence of energy solutions for a Schrödinger-Maxwell system with a variable exponent on a bounded domain $\mathcal {D}\subset \mathbb {R}^N$ $D \subset R^{N}$ ( $N\ge 2$ $N \geq 2$ ), where the p(x)-Laplacian couples the unknown functions u and v. Under the assumptions that $f\in L^{m(\cdot )}(\mathcal {D})$ $f \in L^{m (\cdot)} (D)$ with $m(x)>N/p(x)$ $m (x) > N / p (x)$ and that the continuous exponent satisfies $p(\cdot )>1$ $p (\cdot) > 1$ , a variational approach is employed to establish the existence of weak solutions $(u,v)\in \left( W_{0}^{1,p(\cdot )}(\mathcal {D})\right) ^2$ $(u, v) \in {(W_{0}^{1, p (\cdot)}, (D))}^{2}$ for the considered system. Furthermore, it is shown that this solution is a saddle point of the associated energy functional.