By using the Ljusternik-Schnirelmann category and variational method, we study the existence, multiplicity and concentration of solutions to the fractional Schrödinger equation with potentials competition as follows <Equation ID="Equ1"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="10473_2025_309_Article_Equ1.gif" Format="GIF" Height="28" Rendition="HTML" Resolution="72" Type="Linedraw" Width="342" /> </MediaObject> <EquationSource Format="TEX">$\varepsilon^{N}(-\Delta)_{N/s}^{s}u+V(x)|u|^{{{N}\over{s}}-2}u=Q(x)h(u)\,\,\text{in}\,\, \mathbb{R}^{N},$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msup> <mi>ε</mi> <mrow> <mi>N</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">Δ</mi> <msubsup> <mo stretchy="false">)</mo> <mrow> <mi>N</mi> <mrow> <mo>/</mo> </mrow> <mi>s</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msubsup> <mi>u</mi> <mo>+</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow> <mo stretchy="false">∣</mo> </mrow> <mi>u</mi> <msup> <mrow> <mo stretchy="false">∣</mo> </mrow> <mrow> <mrow> <mfrac> <mrow> <mi>N</mi> </mrow> <mrow> <mi>s</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> <mo>=</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mtext>in</mtext> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>N</mi> </mrow> </msup> <mo>,</mo> </math></EquationSource> </Equation> where ε > 0 is a parameter, s ∈ (0, 1), 2 ≤ p < +∞ and N = ps. The nonlinear term h is a differentiable function with exponential critical growth, the absorption potential V and the reaction potential Q are continuous functions.

Multiplicity and concentration of solutions to a fractional $N\over{S}$-Laplacian problem with exponential critical growth and potentials competition

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发表日期：

2025年2月17日

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Wei Chen,
Chao Ji,
Nguyen Van Thin

摘要

By using the Ljusternik-Schnirelmann category and variational method, we study the existence, multiplicity and concentration of solutions to the fractional Schrödinger equation with potentials competition as follows $\varepsilon^{N}(-\Delta)_{N/s}^{s}u+V(x)|u|^{{{N}\over{s}}-2}u=Q(x)h(u)\,\,\text{in}\,\, \mathbb{R}^{N},$ $ε^{N} (- Δ)_{N / s}^{s} u + V (x) ∣ u ∣^{\frac{N}{s} - 2} u = Q (x) h (u) in R^{N},$ where ε > 0 is a parameter, s ∈ (0, 1), 2 ≤ p < +∞ and N = ps. The nonlinear term h is a differentiable function with exponential critical growth, the absorption potential V and the reaction potential Q are continuous functions.

Multiplicity and concentration of solutions to a fractional \(N\over{S}\)-Laplacian problem with exponential critical growth and potentials competition

摘要