Existence results for Schrödinger–Choquard–Kirchhoff equations involving the fractional p-Laplacian

By Patrizia Pucci, Mingqi Xiang, Binlin Zhang. Advances in Calculus of Variations, 2019

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Abstract The paper is concerned with existence of nonnegative solutions of a Schrödinger–Choquard–Kirchhoff-type fractional p-equation. As a consequence, the results can be applied to the special case ( a + b ⁢ ∥ u ∥ s p ⁢ ( θ - 1 ) ) ⁢ [ ( - Δ ) p s ⁢ u + V ⁢ ( x ) ⁢ | u | p - 2 ⁢ u ] = λ ⁢ f ⁢ ( x , u ) + ( ∫ ℝ N | u | p μ , s * | x - y | μ ⁢ 𝑑 y ) ⁢ | u | p μ , s * - 2 ⁢ u   in ⁢ ℝ N , (a+b\|u\|_{s}^{p(\theta-1)})[(-\Delta)^{s}_{p}u+V(x)|u|^{p-2}u]=\lambda f(x,u)% +\Bigg{(}\int_{\mathbb{R}^{N}}\frac{|u|^{p_{\mu,s}^{*}}}{|x-y|^{\mu}}\,dy% \Biggr{)}|u|^{p_{\mu,s}^{*}-2}u\quad\text{in }\mathbb{R}^{N}, where ∥ u ∥ s = ( ∬ ℝ 2 ⁢ N | u ⁢ ( x ) - u ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y + ∫ ℝ N V ⁢ ( x ) ⁢ | u | p ⁢ 𝑑 x ) 1 p , \|u\|_{s}=\Bigg{(}\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}% \,dx\,dy+\int_{\mathbb{R}^{N}}V(x)|u|^{p}\,dx\Biggr{)}^{\frac{1}{p}}, a , b ∈ ℝ 0 + {a,b\in\mathbb{R}^{+}_{0}} , with a + b > 0 {a+b>0} , λ > 0 {\lambda>0} is a parameter, s ∈ ( 0 , 1 ) {s\in(0,1)} , N > p ⁢ s {N>ps} , θ ∈ [ 1 , N / ( N - p ⁢ s ) ) {\theta\in[1,N/(N-ps))} , ( - Δ ) p s {(-\Delta)^{s}_{p}} is the fractional p-Laplacian, V : ℝ N → ℝ + {V:\mathbb{R}^{N}\rightarrow\mathbb{R}^{+}} is a potential function, 0 < μ < N {0<\mu<N} , p μ , s * = ( p ⁢ N - p ⁢ μ / 2 ) / ( N - p ⁢ s ) {p_{\mu,s}^{*}=(pN-p\mu/2)/(N-ps)} is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality, and f : ℝ N × ℝ → ℝ {f:\mathbb{R}^{N}\times\mathbb{R}\rightarrow\mathbb{R}} is a Carathéodory function. First, via the Mountain Pass theorem, existence of nonnegative solutions is obtained when f satisfies superlinear growth conditions and λ is large enough. Then, via the Ekeland variational principle, existence of nonnegative solutions is investigated when f is sublinear at infinity and λ is small enough. More intriguingly, the paper covers a novel feature of Kirchhoff problems, which is the fact that the parameter a can be zero. Hence the results of the paper are new even for the standard stationary Kirchhoff problems.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/17bdf05fcef852ad1581eacc75e6baadd178a675

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In Applied Mathematics

  1. Binlin Zhang
  2. Sitong Chen
  3. Xianhua Tang
  4. Patrizia Pucci
  5. Sihua Liang
  6. Vicentiu D. Rădulescu
  7. Mingqi Xiang
  8. A. Fiscella

In Computational Theory and Mathematics

  1. Binlin Zhang
  2. Sitong Chen
  3. Xianhua Tang
  4. Patrizia Pucci
  5. Sihua Liang
  6. Mingqi Xiang
  7. Vicentiu D. Rădulescu
  8. A. Fiscella

In Mathematical Physics

  1. Binlin Zhang
  2. Sitong Chen
  3. Xianhua Tang
  4. Sihua Liang
  5. Patrizia Pucci
  6. Mingqi Xiang
  7. Vicentiu D. Rădulescu
  8. A. Fiscella

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