We prove existence of solutions for a semilinear system of Schrödinger–Maxwell type with an exponential nonlinearity: <Equation ID="Equ29"> <EquationSource Format="TEX">$ \left\{ \begin{array}{l} u\in W_0^{1,2}(\Omega ):\ -\textrm{div}(M(x)Du) + \psi \,(\textrm{e}^{|u|}-1)\,\frac{u}{|u|} = f(x), \\ 0 \le \psi \in W_0^{1,2}(\Omega ):\ -\textrm{div}(M(x)D\psi ) = \textrm{e}^{|u|}-|u|-1, \end{array} \right. $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <mi>u</mi> <mo>∈</mo> <msubsup> <mi>W</mi> <mn>0</mn> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mspace width="4pt" /> <mo>-</mo> <mtext>div</mtext> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>D</mi> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>ψ</mi> <mspace width="0.166667em" /> <mrow> <mo stretchy="false">(</mo> <msup> <mtext>e</mtext> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> </msup> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mspace width="0.166667em" /> <mfrac> <mi>u</mi> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> </mfrac> <mo>=</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mn>0</mn> <mo>≤</mo> <mi>ψ</mi> <mo>∈</mo> <msubsup> <mi>W</mi> <mn>0</mn> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mspace width="4pt" /> <mo>-</mo> <mtext>div</mtext> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>D</mi> <mi>ψ</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mtext>e</mtext> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> </msup> <mo>-</mo> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mo>-</mo> <mn>1</mn> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </math></EquationSource> </Equation>where f(x) is such that <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$|f(x)|\,\log (1 + |f(x)|)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">|</mo> <mspace width="0.166667em" /> <mo>log</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mo stretchy="false">|</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">|</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> belongs to <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L^{1}(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Solutions will be saddle points of a suitable functional.

A semilinear system of Schrödinger–Maxwell type with exponential nonlinearities

期刊论文
发表日期：

2026年4月21日

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Lucio Boccardo,
Luigi Orsina

摘要

We prove existence of solutions for a semilinear system of Schrödinger–Maxwell type with an exponential nonlinearity: $ \left\{ \begin{array}{l} u\in W_0^{1,2}(\Omega ):\ -\textrm{div}(M(x)Du) + \psi \,(\textrm{e}^{|u|}-1)\,\frac{u}{|u|} = f(x), \\ 0 \le \psi \in W_0^{1,2}(\Omega ):\ -\textrm{div}(M(x)D\psi ) = \textrm{e}^{|u|}-|u|-1, \end{array} \right. $ $\{\begin{matrix} u \in W_{0}^{1, 2} (Ω) : - div (M (x) D u) + ψ (e^{| u |} - 1) \frac{u}{| u |} = f (x), \\ 0 \leq ψ \in W_{0}^{1, 2} (Ω) : - div (M (x) D ψ) = e^{| u |} - | u | - 1, \end{matrix})$ where f(x) is such that $|f(x)|\,\log (1 + |f(x)|)$ $| f (x) | log (1 + | f (x) |)$ belongs to $L^{1}(\Omega )$ $L^{1} (Ω)$ . Solutions will be saddle points of a suitable functional.