scholarly journals Existence of solutions for $p$-Laplacian equation with electromagnetic fields and critical nonlinearity

2015 ◽  
Vol 22 (4) ◽  
pp. 633-653
Author(s):  
Long Fei ◽  
Hongyan Zhang ◽  
Yueqiang Song
Keyword(s):  
Electromagnetic Fields ◽  
Existence Of Solutions ◽  
Critical Nonlinearity ◽  
Laplacian Equation

scholarly journals Existence of Solutions for thep(x)-Laplacian Problem with the Critical Sobolev-Hardy Exponent

2012 ◽  
Vol 2012 ◽  
pp. 1-17
Author(s):  
Yu Mei ◽  
Fu Yongqiang ◽  
Li Wang
Keyword(s):  
Bounded Domain ◽  
Existence Of Solutions ◽  
Growth Conditions ◽  
Concentration Compactness ◽  
Laplacian Equation

This paper deals with thep(x)-Laplacian equation involving the critical Sobolev-Hardy exponent. Firstly, a principle of concentration compactness inW01,p(x)(Ω)space is established, then by applying it we obtain the existence of solutions for the followingp(x)-Laplacian problem:-div (|∇u|p(x)-2∇u)+|u|p(x)-2u=(h(x)|u|ps*(x)-2u/|x|s(x))+f(x,u),  x∈Ω,  u=0,  x∈∂Ω,whereΩ⊂ℝNis a bounded domain,0∈Ω,1<p-≤p(x)≤p+<N, andf(x,u)satisfiesp(x)-growth conditions.

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