Nonlinear elliptic equations on expanding symmetric domains. (English) Zbl 0944.35026

The authors consider the problem of the existence of nonradial solutions for the semilinear elliptic equation \[ -\Delta u + u = u^p , \;u > 0, \quad \text{in } \Omega_a ,\qquad u = 0 \quad \text{on }\partial \Omega_a \] on expanding (as \(a \rightarrow \infty\)) annulus \[ \Omega_a = \{x \in {\mathbb R}^n : a < |x|< a + 1 \}. \] They introduce a new approach and show how to construct solutions that are obtained as local minimizers of the energy. Their method enables them to get some qualitative properties of solutions such as the shape and the exact symmetry of solutions. For instance, they prove that all solutions obtained are multi-bump solutions with a discrete number of bumps. Under certain conditions they can construct nonradial solutions with prescribed symmetry. They also indicate how the results can be extended to more general domains and more general equations.


35J65 Nonlinear boundary value problems for linear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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[1] Brezis, H.; Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical exponents, Comm. pure appl. math, 36, 437-477, (1983) · Zbl 0541.35029
[2] Byeon, J., Existence of many nonequivalent nonradial positive solutions of semilinear elliptic equations on three-dimensional annuli, J. differential equations, 136, 136-165, (1997) · Zbl 0878.35043
[3] Coffman, C.V., A nonlinear boundary value problem with many positive solutions, J. differential equations, 54, 429-437, (1984) · Zbl 0569.35033
[4] Esteban, M.J., Nonlinear elliptic problems in strip-like domains: symmetry of positive vortex rings, Nonlinear anal., 7, 365-379, (1983) · Zbl 0513.35035
[5] Gidas, B.; Ni, W.-M.; Nirenberg, L., Symmetry and related properties via the maximum principle, Comm. math. phys., 68, 209-243, (1979) · Zbl 0425.35020
[6] Grove, L.C.; Benson, C.T., Finite reflection groups, (1985), Springer-Verlag New York · Zbl 0579.20045
[7] Husain, T., Introduction to topological groups, (1966), Saunders Philadelphia · Zbl 0136.29402
[8] Kawohl, B., Rearrangements and convexity of level sets in P.D.E, Lecture notes in math., (1985), Springer-Verlag Berlin
[9] Li, Y.Y., Existence of many positive solutions of semilinear elliptic equations on annulus, J. differential equations, 83, 348-367, (1990) · Zbl 0748.35013
[10] Lin, S.-S., Existence of many positive nonradial solutions for nonlinear elliptic equations on an annulus, J. differential equations, 103, 338-349, (1993) · Zbl 0803.35053
[11] Lin, S.-S., Asymptotic behavior of positive solutions to semilinear elliptic equations on expanding annuli, J. differential equations, 120, 255-288, (1995) · Zbl 0839.35039
[12] Lions, P.L., The concentration-compactness principle in the calculus of variations. the locally compact case, Anal. nonlinéaire, 1, 109-145, (1984) · Zbl 0541.49009
[13] Lions, P.L., Solutions of Hartree-Fock equations for Coulomb systems, Comm. math. phys., 109, 33-97, (1987) · Zbl 0618.35111
[14] Maier-Paape, S.; Schmitt, K.; Wang, Z.-Q., On Neumann problems for semilinear elliptic equations with critical nonlinearity: existence and symmetry of multi-peaked solutions, Comm. partial differential equations, 22, 1493-1527, (1997) · Zbl 0895.35040
[15] Mizoguchi, N.; Suzuki, T., Semilinear elliptic equations on annuli in three and higher dimensions, Houston J. math., 1, 199-215, (1996) · Zbl 0862.35036
[16] Ni, W.-M.; Takagi, I., On the shape of least-energy solutions to a semilinear Neumann problem, Comm. pure appl. math., 45, 819-851, (1991) · Zbl 0754.35042
[17] Palais, R., The principle of symmetric criticality, Comm. math. phys., 69, 19-30, (1979) · Zbl 0417.58007
[18] Struwe, M., Variational methods, (1990), Springer-Verlag New York
[19] Suzuki, T., Radial and nonradial solutions for semilinear elliptic equations on circular domains, (), 153-174
[20] Wang, Z.-Q., Construction of multi-peaked solutions for a nonlinear Neumann problem with critical exponent in symmetric domains, Nonlinear anal., 27, 1281-1306, (1996) · Zbl 0862.35040
[21] Z.-Q. Wang, Existence and symmetry of multi-bump solutions for nonlinear Schrödinger equations, to appear in, J. Differential Equations.
[22] Willem, M., Minimax theorems, (1996), Birkhäuser Boston · Zbl 0856.49001
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