Multiple positive solutions to critical p-Laplacian equations with vanishing potential

作者：Guo, Lun; Li, Qi^*

来源：Zeitschrift für Angewandte Mathematik und Physik, 2021, 72(4): 167.

DOI：10.1007/s00033-021-01598-4

• p-Laplacian equation
• Critical Sobolev exponent
• Vanishing potential
• Positive solutions
• Ljusternik-Schnirelman theory

This paper deals with the following p-Laplacian equation @@@ -epsilon(p)Delta(p)u + V(x)vertical bar u vertical bar(p-2)u = vertical bar u vertical bar(p)*(-2)u, u is an element of D-1,D-p(R-N), @@@ where p is an element of (1, N), p-Laplacian operator Delta(p):=div(vertical bar del u vertical bar(p-2)del u), p* = Np/(N - p), epsilon is a positive parameter, V(x) is an element of L-N/p(R-N) boolean AND L-loc(infinity)(R-N) and V(x) is assumed to be zero in some region of R-N, which means it is of the vanishing potential case. In virtue of Ljusternik-Schnirelman theory of critical points, we succeed in proving the multiplicity of positive solutions. This result generalizes the result for semilinear Schrodinger equation by Chabrowski and Yang (Port. Math. 57 (2000), 273-284) to p-Laplacian equation.

• 单位
  复旦大学