Opuscula Math. 34, no. 2 (2014), 271-290
http://dx.doi.org/10.7494/OpMath.2014.34.2.271

Opuscula Mathematica

# On a singular nonlinear Neumann problem

Jan Chabrowski

Abstract. We investigate the solvability of the Neumann problem involving two critical exponents: Sobolev and Hardy-Sobolev. We establish the existence of a solution in three cases: $$\text{(i)}\;\ 2\lt p+1\lt 2^*(s),$$ $$\text{(ii)}\;\ p+1=2^*(s)$$ and $$\text{(iii)}\;\ 2^*(s)\lt p+1 \leq 2^*,$$ where $$2^*(s)=\frac{2(N-s)}{N-2},$$ $$0\lt s\lt 2,$$ and $$2^*=\frac{2N}{N-2}$$ denote the critical Hardy-Sobolev exponent and the critical Sobolev exponent, respectively.

Keywords: Neumann problem, critical Sobolev exponent, Hardy-Sobolev exponent Neumann problem.

Mathematics Subject Classification: 35B33, 35J20, 35J65.

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• Jan Chabrowski
• University of Queensland, Department of Mathematics, St. Lucia 4072, Qld, Australia