In this paper, by using variational methods, the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {Z}_{2}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-genus, and the Moser iteration technique, we study the following quasilinear Schrödinger equation: <Equation ID="Equ21"> <EquationSource Format="TEX">$\begin{aligned} - \Delta u + V(x)u + \tau \Delta \!\left( \sqrt{1 + u^2} \right) \frac{u}{2\sqrt{1 + u^2}} = W(x)f(x, u), \quad x \in \mathbb {R}^N, \end{aligned}$</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>τ</mi> <mi mathvariant="normal">Δ</mi> <mspace width="-0.166667em" /> <mfenced close=")" open="("> <msqrt> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mfenced> <mfrac> <mi>u</mi> <mrow> <mn>2</mn> <msqrt> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> <mo>=</mo> <mi>W</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>x</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$N \ge 3 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ \tau \ge 2 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>τ</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, and the nonlinearity f(x, t) is sublinear in a neighborhood of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ t = 0.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation>

A new class of quasilinear Schrödinger equations with sublinear nonlinearity

期刊论文
发表日期：

2025年11月17日

查看全文

Safa Bridaa,
Abderrazek B. Hassine

摘要

In this paper, by using variational methods, the $\mathbb {Z}_{2}$ $Z_{2}$ -genus, and the Moser iteration technique, we study the following quasilinear Schrödinger equation: $\begin{aligned} - \Delta u + V(x)u + \tau \Delta \!\left( \sqrt{1 + u^2} \right) \frac{u}{2\sqrt{1 + u^2}} = W(x)f(x, u), \quad x \in \mathbb {R}^N, \end{aligned}$ $\begin{matrix} - Δ u + V (x) u + τ Δ (\sqrt{1 + u^{2}}) \frac{u}{2 \sqrt{1 + u^{2}}} = W (x) f (x, u), x \in R^{N}, \end{matrix}$ where $N \ge 3 $ $N \geq 3$ , $ \tau \ge 2 $ $τ \geq 2$ , and the nonlinearity f(x, t) is sublinear in a neighborhood of $ t = 0.$ $t = 0 .$