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Existence and multiplicity of solutions for fractional Schrödinger-p-Kirchhoff equations in ℝ N
Forum Mathematicum ( IF 0.8 ) Pub Date : 2024-03-25 , DOI: 10.1515/forum-2023-0385 Huo Tao 1 , Lin Li 1 , Patrick Winkert 2
Forum Mathematicum ( IF 0.8 ) Pub Date : 2024-03-25 , DOI: 10.1515/forum-2023-0385 Huo Tao 1 , Lin Li 1 , Patrick Winkert 2
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This paper concerns the existence and multiplicity of solutions for a nonlinear Schrödinger–Kirchhoff-type equation involving the fractional p -Laplace operator in ℝ N {\mathbb{R}^{N}} . Precisely, we study the Kirchhoff-type problem ( a + b ∬ ℝ 2 N | u ( x ) - u ( y ) | p | x - y | N + s p d x d y ) ( - Δ ) p s u + V ( x ) | u | p - 2 u = f ( x , u ) in ℝ N , \Biggl{(}a+b\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\,% \mathrm{d}x\,\mathrm{d}y\Biggr{)}(-\Delta)^{s}_{p}u+V(x)|u|^{p-2}u=f(x,u)\quad% \text{in }\mathbb{R}^{N}, where a , b > 0 {a,b>0} , ( - Δ ) p s {(-\Delta)^{s}_{p}} is the fractional p -Laplacian with 0 < s < 1 < p < N s {0<s<1<p<\frac{N}{s}} , V : ℝ N → ℝ {V\colon\mathbb{R}^{N}\to\mathbb{R}} and f : ℝ N × ℝ → ℝ {f\colon\mathbb{R}^{N}\times\mathbb{R}\to\mathbb{R}} are continuous functions while V can have negative values and f fulfills suitable growth assumptions. According to the interaction between the attenuation of the potential at infinity and the behavior of the nonlinear term at the origin, using a penalization argument along with L ∞ {L^{\infty}} -estimates and variational methods, we prove the existence of a positive solution. In addition, we also establish the existence of infinitely many solutions provided the nonlinear term is odd.
中文翻译:
ℝ N 中分数阶薛定谔-p-基尔霍夫方程解的存在性和多重性
本文关注涉及分数的非线性薛定谔-基尔霍夫型方程解的存在性和多重性p - 拉普拉斯算子 ℝ 氮 {\mathbb{R}^{N}} 。准确地说,我们研究的是基尔霍夫型问题 ( A + 乙 ∬ ℝ 2 氮 | 你 ( X ) - 你 ( y ) | p | X - y | 氮 + s p d X d y ) ( - Δ ) p s 你 + V ( X ) | 你 | p - 2 你 = F ( X , 你 ) 在 ℝ 氮 , \Biggl{(}a+b\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|xy|^{N+sp} }\,% \mathrm{d}x\,\mathrm{d}y\Biggr{)}(-\Delta)^{s}_{p}u+V(x)|u|^{p-2 }u=f(x,u)\quad% \text{在 }\mathbb{R}^{N}, 在哪里 A , 乙 > 0 {a,b>0} , ( - Δ ) p s {(-\Delta)^{s}_{p}} 是分数p -拉普拉斯算子 0 < s < 1 < p < 氮 s {0<s<1<p<\frac{N}{s}} , V : ℝ 氮 → ℝ {V\colon\mathbb{R}^{N}\to\mathbb{R}} 和 F : ℝ 氮 × ℝ → ℝ {f\colon\mathbb{R}^{N}\times\mathbb{R}\to\mathbb{R}} 是连续函数,而V 可以有负值并且F 满足适当的增长假设。根据无穷远处势能的衰减与原点处非线性项的行为之间的相互作用,使用惩罚参数 L 无穷大 {L^{\infty}} -估计和变分方法,我们证明了正解的存在。此外,如果非线性项是奇数,我们还证明存在无限多个解。
更新日期:2024-03-25
中文翻译:
ℝ N 中分数阶薛定谔-p-基尔霍夫方程解的存在性和多重性
本文关注涉及分数的非线性薛定谔-基尔霍夫型方程解的存在性和多重性