On a p-Laplace equation with multiple critical nonlinearities

Abstract

Using the Mountain-Pass Theorem of Ambrosetti and Rabinowitz we prove thatThe technique is based on the existence of extremals of some Hardy-Sobolev type embeddings of independent interest. We also show that if u ∈ D p 1 (R n ) is a weak solution in R n of −∆pu − µ|x| −p |u| p−2 u = |x| −s |u| p ⋆ (s)−2 u + |u| q−2 u, then u ≡ 0 when either 1 < q < p ⋆ , or q > p ⋆ and u is also of class L ∞ loc (R n \ {0}).

Multiple positive solutions for a class of quasi-linear elliptic equations involving concave-convex nonlinearities and Hardy terms

“…Such kind of problem with critical exponents and nonnegative weight functions has been extensively studied by many authors. We refer, e.g., in bounded domains and for p = 2 to [4][5][6] and for p >1 to [7][8][9][10][11], while in ℝ N and for p = 2 to [12,13], and for p >1 to [3,[14][15][16][17], and the references therein.…”

Existence and multiplicity of positive solutions for p-Laplacian elliptic equations

“…China. 2 School of Mathematical Sciences, University of Jinan, Jinan, Shandong Province 250022, P.R. China.…”

Five Nontrivial Solutions of p-Laplacian Problems Involving Critical Exponents and Singular Cylindrical Potential

“…If the weight functions f ≡ h ≡ 1, the authors Ambrosetti-Brezis-Cerami [1] have investigated equation They found that there exists such that equation admits at least two positive solutions for and has a positive solution for but no positive solution exists for For more general results, were done by de Figueiredo-Grossez-Ubilla [2], Wu [3], Cao etal. [4], Filippucci et al [5], Xuan et al [6], Guo and Niu [7] and the references therein.…”