This paper establishes an arbitrary number of solutions for the following Schrödinger–Bopp–Podolsky system with lack of symmetry: <Equation ID="Equ20"> <EquationSource Format="TEX">$ {\left\{ \begin{array}{ll} -\Delta u + V(x) u + \lambda \phi u = f(u), & \text {in} \mathbb {R}^3, \\ -\Delta \phi + a^2\Delta ^2\phi = 4\pi u^2, & \text {in} \mathbb {R}^3, \end{array}\right. } $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>+</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>u</mi> <mo>+</mo> <mi>λ</mi> <mi>ϕ</mi> <mi>u</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mtext>in</mtext> <mspace width="3.33333pt" /> <mspace width="3.33333pt" /> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>ϕ</mi> <mo>+</mo> <msup> <mi>a</mi> <mn>2</mn> </msup> <msup> <mi mathvariant="normal">Δ</mi> <mn>2</mn> </msup> <mi>ϕ</mi> <mo>=</mo> <mn>4</mn> <mi>π</mi> <msup> <mi>u</mi> <mn>2</mn> </msup> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mtext>in</mtext> <mspace width="3.33333pt" /> <mspace width="3.33333pt" /> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$a\ne 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>≠</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and the nonlinear function f(t) satisfies the superlinear assumptions only near <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$t=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. Using the essential values theory and a new truncation method, we show that for any <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$k \in \mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation>, there exists <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\lambda _k < 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mi>k</mi> </msub> <mo><</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> such that for all <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\lambda \le \lambda _k$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>≤</mo> <msub> <mi>λ</mi> <mi>k</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, the system admits at least k distinct nontrivial solutions. In this paper, we successfully eliminated the Ambrosetti–Rabinowitz condition, growth conditions at infinity and symmetry condition on the nonlinear function.

An arbitrary number of solutions for Schrödinger–Bopp–Podolsky systems with lack of symmetry

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发表日期：

2026年4月1日

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Chen Huang,
Chen Dong,
Xian Zhang

摘要

This paper establishes an arbitrary number of solutions for the following Schrödinger–Bopp–Podolsky system with lack of symmetry: $ {\left\{ \begin{array}{ll} -\Delta u + V(x) u + \lambda \phi u = f(u), & \text {in}~~ \mathbb {R}^3, \\ -\Delta \phi + a^2\Delta ^2\phi = 4\pi u^2, & \text {in}~~ \mathbb {R}^3, \end{array}\right. } $ $\{\begin{matrix} - Δ u + V (x) u + λ ϕ u = f (u), & in R^{3}, \\ - Δ ϕ + a^{2} Δ^{2} ϕ = 4 π u^{2}, & in R^{3}, \end{matrix})$ where $a\ne 0$ $a \neq 0$ and the nonlinear function f(t) satisfies the superlinear assumptions only near $t=0$ $t = 0$ . Using the essential values theory and a new truncation method, we show that for any $k \in \mathbb {N}$ $k \in N$ , there exists $\lambda _k < 0$ $λ_{k} < 0$ such that for all $\lambda \le \lambda _k$ $λ \leq λ_{k}$ , the system admits at least k distinct nontrivial solutions. In this paper, we successfully eliminated the Ambrosetti–Rabinowitz condition, growth conditions at infinity and symmetry condition on the nonlinear function.