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Existence of solutions with prescribed norm for semilinear elliptic equations. (English) Zbl 0877.35091

The nonlinear eigenvalue problem \[ -\Delta u(x) - g(u(x)) = \lambda u(x),\quad\lambda \in \mathbb{R},\;x \in \mathbb{R}^N \] is studied. It is supposed that \(g:\mathbb{R} \to \mathbb{R}\) is continuous, odd, and such that \[ \alpha \int_0^s g(t) dt \leq g(s)s \leq \beta \int_0^s g(t) dt \] with some \((2N+4)/N<\alpha\leq\beta< (2N)/(N-2)\) if \(N\geq 3\) or \((2N+4)/N<\alpha\leq\beta\) if \(n=1,2\). A certain mild additional assumption is added in the case \(N=1\). It is proved that for any \(c>0\) there exists a weak solution \(u_c \in H^1(\mathbb{R}^N),\;\lambda_c \in \mathbb{R}\) satisfying \(|u|_{L^2}= c,\;\lambda_c <0\). The proof is based on a minimax approach used for the corresponding functional. Further, the dependence of \(|\nabla u_c|_{L^2}\) and \(\lambda_c\) on \(c\) is described. Particularly, it follows \(|\nabla u_c|_{L^2} \to +\infty,\;\lambda_c \to -\infty\) as \(c \to 0\) and \(|\nabla u_c|_{L^2} \to 0,\;\lambda_c \to 0\) as \(c \to +\infty\).
Reviewer: M.Kučera (Praha)

MSC:

35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
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[1] Ambrosetti, A.; Bertotti, M.L., Homoclinics for second order convervative systems. partial differential equations and related subjects, () · Zbl 0804.34046
[2] Ambrosetti, A.; Struwe, M., Existence of steady vortex rings in an ideal fluid, Arch. rat. mech. anal., 108, 97-109, (1989) · Zbl 0694.76012
[3] Benci, V.; Cerami, G., Positive solutions of some nonlinear elliptic problems in exterior domains, Arch. rat. mech. anal., 99, 283-300, (1987) · Zbl 0635.35036
[4] Berestycki, H.; Lions, P.L., Nonlinear scalar field equations I, Arch. rat. mech. anal., 82, 313-346, (1983)
[5] Berestycki, H.; Lions, P.L., Nonlinear scalar field equations II, Arch. rat. mech. anal., 82, 347-376, (1983)
[6] Brezis, H.; Kato, T., Remarks on the schrodinger operator with singular complex potentials, J. mat. pures appli., 58, 137-151, (1979) · Zbl 0408.35025
[7] Buffoni, B., Un problème variationnel fortement indéfini sans compacité, ()
[8] Buffoni B., Private communication, April 1992.
[9] Buffoni, B.; Jeanjean, L., Minimax characterization of solutions for a semilinear elliptic equation, Ann. inst. H. Poincaré, anal. non-lin., 10, 4, 377, (1993) · Zbl 0828.35013
[10] Jeanjean, L., Approche minimax des solutions d’une équation semi-linéaire elliptique en l’absence de compacité, ()
[11] Lions, J.L., Problèmes aux limites dans LES équations aux dérivés partielles, (1962), Presses de l’univ.
[12] Lions, P.L., The concentration-compactness principle in the calculus of variations, part 1, Ann. inst. H. Poincaré, anal, non-lin., 1, 109-145, (1984) · Zbl 0541.49009
[13] Lions, P.L., The concentration-compactness principle in the calculus of variations, part 2, Ann. inst. H. Poincaré, anal, non-lin., 1, 223-283, (1984) · Zbl 0704.49004
[14] Krasnoselskii, M.A., Topological methods in the theory of nonlinear integral operators, (1964), Pergamon Press de Montréal
[15] Mawhin, J.; Willem, M., Critical point theory and Hamiltonian systems, Appli. math. sci., 74, (1989) · Zbl 0676.58017
[16] Pohozaev, S., Eigenfunctions of the equations δu + λƒ (u) = 0, Soviet math. dokl., 6, 1408-1411, (1965) · Zbl 0141.30202
[17] Rabinowitz, P.H., Minimax methods in critical point theory with applications to differential equations, () · Zbl 0152.10003
[18] Strauss, W.A., Existence of solitary waves in higher dimensions, Comm. math. phys., 55, 149-162, (1977) · Zbl 0356.35028
[19] Struwe, M., ()
[20] Stuart, C.A., Bifurcation for variational problems when the linearization has no eigenvalues, J. funct. anal., 38, 169-187, (1980) · Zbl 0458.47048
[21] Stuart, C.A., Bifurcation from the continuous spectrum in L2-theory of elliptic equations on{\bfr}n, ()
[22] Stuart, C.A., Bifurcation in L^{p} ({\bfr}N) for a semilinear elliptic equation, (), 511-541, 3 · Zbl 0673.35005
[23] Stuart, C.A., Bifurcation from the essential spectrum for some non-compact nonlinearities, Math. appl. sci., 11, 525-542, (1989) · Zbl 0678.58013
[24] Tanaka K., Private communication, May 1993.
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