Positive Solutions for a Class of Fractional p-Laplacian Equation with Critical Sobolev Exponent and Decaying Potentials

• Published in: Acta Mathematicae Applicatae Sinica Vol. 38; no. 2; pp. 463 - 483
• Main Authors: Li, Na, He, Xiao-ming
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.04.2022; Springer Nature B.V
• Edition: English series
• Subjects: Applications of Mathematics; Continuity (mathematics); Laplace equation; Math Applications in Computer Science; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Theoretical; 35J62; Variational methods; 35J50; Mountain Pass Theorem; 35B65; Fractional; Critical growth; Vanishing potential; Laplacian
• ISSN: 0168-9673; 1618-3932
• DOI: 10.1007/s10255-022-1090-8

In this paper, we study the existence of positive solution for the p -Laplacian equations with fractional critical nonlinearity { ( − Δ ) p s u + V ( x ) | u | p − 2 u = K ( x ) f ( u ) + P ( x ) | u | p s ∗ − 2 u , x ∈ ℝ N , u ∈ D s , p ( ℝ N ) , where s ∈ ( 0 , 1 ) , p s ∗ = N p N − s p , N > s p , p > 1 and V ( x ), K ( x ) are positive continuous functions which vanish at infinity, f is a function with a subcritical growth, and P ( x ) is bounded, nonnegative continuous function. By using variational method in the weighted spaces, we prove the above problem has at least one positive solution.