## 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

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 35J20 Variational methods for second-order elliptic equations 58E30 Variational principles in infinite-dimensional spaces
Full Text:

### References:

 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.