A nonlinear boundary value problem with many positive solutions. (English) Zbl 0569.35033

The nonlinear problem \(-\Delta u+u=u^{2N+1}\) in G, \(u=0\) on \(\partial G\), where G is the annulus \(\{(x,y):\quad r^ 2<x^ 2+y^ 2<(r+c)^ 2\}\) and N is a positive integer, is considered in this paper. The author proves the following interesting result: for fixed \(c>0\), the number of rotationally non-equivalent positive solutions of this problem tends to infinity as \(r\to +\infty\). As a corollary it is possible to show that for a fixed annulus the number of (rotationally non-equivalent) positive solutions of the problem \(-\Delta u+m^ 2u=u^{2N+1}\) in G, \(u=0\) on \(\partial G\), tends to infinity as \(m\to \infty\). The proof uses a variational principle for a Rayleigh quotient on subspaces of the Sobolev space \(H^ 1_ 0(G)\) with some symmetry properties.
Reviewer: J.Hernández


35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
58E30 Variational principles in infinite-dimensional spaces
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[1] Brezis, H; Nirenberg, L, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. pure and appl. math., 36, 437-477, (1983) · Zbl 0541.35029
[2] Coffman, C.V, Uniqueness of the ground state solution for δu − u + u3 = 0 and a variational characterization of other solutions, Arch. rational mech. anal., 46, 81-95, (1972) · Zbl 0249.35029
[3] Coffman, C.V, An alternate variational principle for \(Δu − u + ¦u¦\^{}\{−1\}\) sgn u = 0, (), 666-670 · Zbl 0534.35042
[4] Friedman, A, Partial differential equations, (1969), Holt, Rinehart & Winston New York
[5] Gidas, B; Ni, W; Nirenberg, L, Symmetry and related properties via the maximum principle, Comm. math. phys., 68, 209-243, (1979) · Zbl 0425.35020
[6] Lions, P.-L, Symétrie et compacité dans LES espaces de Sobolev, J. funct. anal., 49, 315-334, (1982) · Zbl 0501.46032
[7] Nehari, Z, On a nonlinear differential equation arising in nuclear physics, (), 117-135 · Zbl 0124.30204
[8] Strauss, W.A, Existence of solitary waves in higher dimensions, Comm. math. phys., 55, 149-162, (1977) · Zbl 0356.35028
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