We consider the nonlinear elliptic equation <Equation ID="Equ19"> <EquationSource Format="TEX">$\begin{aligned} -\Delta u + V(x)u = f(u), \qquad u\in D^{1,2}_0(\Omega ), \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>+</mo> <mi>V</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>u</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="2em" /> <mi>u</mi> <mo>∈</mo> <msubsup> <mi>D</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> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>in an exterior domain <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {R}^N$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation>, where V is a scalar potential that decays to zero at infinity and the nonlinearity f is subcritical at infinity and supercritical near the origin. Under weak symmetry assumptions, we provide conditions that guarantee that this problem has a prescribed number of sign-changing solutions. In particular, we show that in dimensions <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$N\ge 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>≥</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation> there are numerous examples of exterior domains with finite symmetries in which the problem has a predetermined number of nodal solutions.

Multiple nodal solutions to a scalar field equation with double-power nonlinearity and zero mass at infinity

期刊论文
出版模式：

开放获取
发表日期：

2026年4月18日

查看全文

Mónica Clapp,
Carlos Culebro

摘要

We consider the nonlinear elliptic equation $\begin{aligned} -\Delta u + V(x)u = f(u), \qquad u\in D^{1,2}_0(\Omega ), \end{aligned}$ $\begin{matrix} - Δ u + V (x) u = f (u), u \in D_{0}^{1, 2} (Ω), \end{matrix}$ in an exterior domain $\Omega $ $Ω$ of $\mathbb {R}^N$ $R^{N}$ , where V is a scalar potential that decays to zero at infinity and the nonlinearity f is subcritical at infinity and supercritical near the origin. Under weak symmetry assumptions, we provide conditions that guarantee that this problem has a prescribed number of sign-changing solutions. In particular, we show that in dimensions $N\ge 4$ $N \geq 4$ there are numerous examples of exterior domains with finite symmetries in which the problem has a predetermined number of nodal solutions.