Symmetry-breakings for semilinear elliptic equations on finite cylindrical domains

Author:
Song-Sun Lin

Journal:
Proc. Amer. Math. Soc. **117** (1993), 803-811

MSC:
Primary 35B05; Secondary 35B32, 35J65, 35P30

MathSciNet review:
1116265

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Abstract: We study the existence and multiplicity of asymmetric positive solutions of a semilinear elliptic equation on finite cylinders with mixed type boundary conditions. By using a Nehari-type variational method, we prove that the numbers of asymmetric positive solutions are increasing without bound when the lengths of cylinders are increasing. On the contrary, by using the blow up technique, we obtain an a priori bound for positive solutions and then prove that all positive solutions must be symmetric when the cylinders are short enough.

**[1]**Antonio Ambrosetti and Paul H. Rabinowitz,*Dual variational methods in critical point theory and applications*, J. Functional Analysis**14**(1973), 349–381. MR**0370183****[2]**Henri Berestycki and Filomena Pacella,*Symmetry properties for positive solutions of elliptic equations with mixed boundary conditions*, J. Funct. Anal.**87**(1989), no. 1, 177–211. MR**1025886**, 10.1016/0022-1236(89)90007-4**[3]**Haïm Brézis and Louis Nirenberg,*Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents*, Comm. Pure Appl. Math.**36**(1983), no. 4, 437–477. MR**709644**, 10.1002/cpa.3160360405**[4]**Giovanna Cerami,*Symmetry breaking for a class of semilinear elliptic problems*, Nonlinear Anal.**10**(1986), no. 1, 1–14. MR**820654**, 10.1016/0362-546X(86)90007-6**[5]**Charles V. Coffman,*A nonlinear boundary value problem with many positive solutions*, J. Differential Equations**54**(1984), no. 3, 429–437. MR**760381**, 10.1016/0022-0396(84)90153-0**[6]**E. N. Dancer,*On nonradially symmetric bifurcation*, J. London Math. Soc. (2)**20**(1979), no. 2, 287–292. MR**551456**, 10.1112/jlms/s2-20.2.287**[7]**B. Gidas and J. Spruck,*A priori bounds for positive solutions of nonlinear elliptic equations*, Comm. Partial Differential Equations**6**(1981), no. 8, 883–901. MR**619749**, 10.1080/03605308108820196**[8]**Yan Yan Li,*Existence of many positive solutions of semilinear elliptic equations on annulus*, J. Differential Equations**83**(1990), no. 2, 348–367. MR**1033192**, 10.1016/0022-0396(90)90062-T**[9]**Song-Sun Lin,*On non-radially symmetric bifurcation in the annulus*, J. Differential Equations**80**(1989), no. 2, 251–279. MR**1011150**, 10.1016/0022-0396(89)90084-3**[10]**Song Sun Lin,*Positive radial solutions and nonradial bifurcation for semilinear elliptic equations in annular domains*, J. Differential Equations**86**(1990), no. 2, 367–391. MR**1064016**, 10.1016/0022-0396(90)90035-N**[11]**-,*Existence of positive nonradial solutions for elliptic equations in annular domains*, Trans. Amer. Math. Soc. (to appear).**[12]**Song-Sun Lin,*Symmetry breaking for semilinear elliptic equations on sectorial domains in 𝑅²*, Proc. Roy. Soc. Edinburgh Sect. A**118**(1991), no. 3-4, 327–353. MR**1121671**, 10.1017/S0308210500029127**[13]**Zeev Nehari,*On a class of nonlinear second-order differential equations*, Trans. Amer. Math. Soc.**95**(1960), 101–123. MR**0111898**, 10.1090/S0002-9947-1960-0111898-8**[14]**W.-M. Ni,*Some aspects of semilinear elliptic equations*, lecture notes, National Tsing Hua University, Hsinchu, Taiwan, May 1987.**[15]**Mythily Ramaswamy and P. N. Srikanth,*Symmetry breaking for a class of semilinear elliptic problems*, Trans. Amer. Math. Soc.**304**(1987), no. 2, 839–845. MR**911098**, 10.1090/S0002-9947-1987-0911098-4**[16]**Joel Smoller and Arthur Wasserman,*Symmetry-breaking for positive solutions of semilinear elliptic equations*, Arch. Rational Mech. Anal.**95**(1986), no. 3, 217–225. MR**853965**, 10.1007/BF00251359**[17]**Joel A. Smoller and Arthur G. Wasserman,*Symmetry-breaking for solutions of semilinear elliptic equations with general boundary conditions*, Comm. Math. Phys.**105**(1986), no. 3, 415–441. MR**848648****[18]**Takashi Suzuki and Ken’ichi Nagasaki,*Lifting of local subdifferentiations and elliptic boundary value problems on symmetric domains. I*, Proc. Japan Acad. Ser. A Math. Sci.**64**(1988), no. 1, 1–4. MR**953750****[19]**-,*On the nonlinear eigenvalue problem*, Trans. Amer. Math. Soc.**309**(1988), 591-608.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1993-1116265-3

Keywords:
Symmetry breaking,
cylinders

Article copyright:
© Copyright 1993
American Mathematical Society