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Conference Equadiff12

REF. DATE:24 July 2009 CREATED/MODIFIED: 15 January 2012 364 views No Comment

I gave the talk “Four solutions for an elliptic equation with critical exponent and singular term” at Equadiff 12, July 20-24, 2009, Brno, Czech Republic.

We show, under some conditions, that the Dirichlet problem

-\Delta u - {\lambda\over {|x|^2}}u = |u|^{2^{\ast}-2}u  + \mu|x|^{\alpha-2}u + f(x) a.e. on \Omega\backslash\{0\} with u\in H_0^1(\Omega),

has four nontrivial solutions where at least one of solutions is sign changing. We assume that 0\in \Omega\subset\mathbb{R}^N (with N\geq 3) is a bounded domain with smoothboundary, 2^{\ast}:=2N/(N-2) is the critical Sobolev exponent, 0\leq \lambda < \Lambda:=((N-2)/2)^2 and f\in L^\infty(\Omega). These results extend some previous works on the literature, as [1-3]. This is a joint work with Jianqing Chen.
[1] N. Hirano and N. Shioji, A multiplicity result including a sign changing solution for an inhomogeneous Neumann problem with critical exponent, Proc. Roy. Soc. Edinburgh 137A(2007) 333-347.
[2] D. Kang and Y. Deng, Multiple solutions for inhomogeneous elliptic problems involving critical Sobolev-Hardy exponents, J. Math. Anal. Appl. 60(2005) 729-753.
[3] G. Tarantello, Multiplicity results for an inhomogeneous Neumann problem with critical exponent, Manuscripta Math. 81(1993) 51-78.

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