. (1) 2c Bahri- Bahri-Coron u = u 4/(N 2) u

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2 . (1) Nehari c (c, 2c) 2c Bahri- Coron Bahri-Lions (2) Hénon u = x α u p α (3) u(x) u(x) + u(x) p = 0... (1) 1 Ω R N f : R R Neumann d 2 u + u = f(u) d > 0 Ω f Dirichlet 2 Ω R N ( ) Dirichlet Bahri-Coron u = u 4/(N 2) u 1 c c, 2c Palais-Smale (c, 2c) 2c 3 Ω R N Dirichlet u λu = u 4/(N 2) u N 5 λ 1 4 R 3 Dirichlet Hénon u = x α u p α (2) 1 R N Dirichlet u + f( x, u) = 0 2 R 2 Dirichlet 2π/n C n

3 Hénon 2π/n (3) Dirichlet u + g( x )u + h( x )u p = 0 Pohožaev Existence of many nonradial positive solutions of the Hénon equation in R 3, Journal of Convex Analysis 22,,, JCA22/jca22.htm 2 Naoki Shioji and Kohtaro Watanabe, Radial symmetry of n-mode positive solutions for semilinear elliptic equations in a disc and its application to the Hénon equation, Topological methods in nonlinear analysis 43, , 2014,, vol-43-1.html 3 Norimichi Hirano and Naoki Shioji, A geometric proof of the existence of two solutions of the Bahri-Coron problem, Springer Proceedings in Mathematics & Statistics 47, , 2013,, / Naoki Shioji and Kohtaro Watanabe, Uniqueness of positive radial solutions of u + g(r)u + h(r)u p = 0 and its applications, 1838, 64 70, 2013,, issue/nels/ AN /ISS ja.html 5 Naoki Shioji and Kohtaro Watanabe, A generalized Pohožaev identity and uniqueness of positive radial solutions of u + g(r)u + h(r)u p = 0, Journal of differential equations 255, , 2013, /j.jde Naoki Shioji, Existence of multiple sign-changing solutions for an asymptotically linear elliptic problem and the topology of the configuration space of the domain, Advances in Differential Equations 17, , 2012,, Norimichi Hirano and Naoki Shioji, Existence of two solutions tor the Bahri- Coron problem in an annular domain with a thin hole, Journal of Functional Analysis 261, , 2012,, /j.jfa Zhijie Chen, Naoki Shioji and Wenming Zou, Ground state and multiple solutions for a critical exponent problem, NoDEA. Nonlinear Differential Equations and Applications 19, , 2012, /s Naoki Shioji and Kohtaro Watanabe, Radial symmetry of positive solutions for scmilinear elliptic equations in the unit ball via elliptic and hyperbolic geometry, Journal of Differential Equations 252, , 2011,, / /j.jde Naoki Shioji, Multiple sign-changing solutions for an asymptotically linear elliptic problem, 1740, , 2011,, issue/nels/ AN /ISS ja.html 11 Naoki Shioji, Multiple sign-changing solutions for a semilinear Neumann problem and the topology of the configuration space of the domain boundary, Calculus of Variations and Partial Differential Equations 38, , 2010,, /s x 12 Yoshihiro Mizuta, Takao Ohno, Tetsu Shimomura and Naoki Shioji, Com-

4 pact embeddings for Sobolev spaces of variable exponents and existence of solutions for nonlinear elliptic problems involving the p(x)-laplacian and its critical exponent, Annales Academiae Scientiarum Fennicae, Series AI, Mathematica 35, , 2010,, Vol35/vol35.html 23 1,, u + ρ u/ρ gu + hu p = 0, 2014, , 2,, 2014, , 3 Naoki Shioji, Uniqueness of positive radial solutions of u + g(r)u + h(r)u p = 0, The international conference on nonlinear analysis and optimization,, , () 4,, 117, , 5,, 571, , 6 Norimichi Hirano and Naoki Shioji, Two solutions for the Bahri-Coron problem, The eighth international conference on nonlinear analysis and convex analysis,, , 7 Naoki Shioji, Uniqueness of positive radial solutions of u + g(r)u + h(r)u p = 0 and its applications, The Asiain mathematical conference 2013,, , () 8,, u + g(r)u+h(r)u p = 0,, , 9, u + g(r)u + h(r)u p = 0,, , 10, Some results on the positive solutions of the Hénon equation, International Workshop on Stationary Problems in Nonlinear Partial Differential Equations, , 11, Uniqueness of positive radial solutions of u + g(r)u + h(r)u p = 0 and its applications, RIMS, , 12,, n Hénon,, , 13, n Hénon,, , 14, Partial radial symmetry of positive solutions for semilinear elliptic equations in a disc and its application to the H enon equation, Spring school on nonlinear partial differential equations, , () 15,, 2011, , 16,, -, 2011, , 17, Two solutions for a Bahri-Coron problem in an annular domain with a thin hole, International Conference on Differ-

5 ential & Difference Equations and Applications, , ( ) 18, Brezis-Nrenberg, , 19,, ( 33 ), , 20, Multiple sign changing solutions for an asymptotically linear elliptic problems, RIMS, , 21, Existence of multiple signchanging solutions for semilinear elliptic equations via the Lusternik-Schnirelmann category, 2009, , 22, Neumann, , 23, Existence of multiple signchanging solutions for a singularly perturbed Neumann problem, Equadiff12, , ( ) (1) (SHIOJI Naoki) (2) (HIRANO Norimichi) (TAMANO Kenichi) (WATANABE Kohtaro)

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