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Dual variational methods in critical point theory and applications. (English) Zbl 0273.49063
Consider the nonlinear elliptic partial differential equation \[ L(u) \equiv -\sum_{i,j=1}^n (a_{ij}(x)u_{x_i})_{x_j} + c(x)u = p(x,u),\quad x\in\Omega,\ u = 0,\ x \in\partial\Omega, \tag{*}\]
where \(\Omega\subset\mathbb R^n\) is a smooth bounded domain. Formally, the critical points of the functional
\[ I(u) = \int_\Omega \left[ \frac12 \sum_{i,j=1}^n (a_{ij}(x)u_{x_i})_{x_j} + c(x)u^2 - P(x,u(x))\right] \,dx, \]
where \(P(x,u)\) is a primitive of \(p(x,u)\), are solutions of (*). The authors construct dual variational methods to enable them to prove the existence and estimate the number of critical points possessed by a real continuously differentiable functional on a real Banach space, and then apply their results to various existence problems for equations of type (*). They also apply them to problems with linear term added, i.e.
\[ L(u) = a(x)u + p(x,u),\quad x\in\Omega;\ u=0,\ x \in\partial\Omega, \]
as well as to nonlinear integral equations of the form
\[ v(x) = \int_\Omega g(x,y)q(y,v(y))\,dy. \]
Reviewer: H. S. P. Grässer

58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
35J20 Variational methods for second-order elliptic equations
Full Text: DOI
[1] Ljusternik, L.A; Schnirelman, L.G, Methodes topologiques dans LES problèmes variationels, Actualites sci. ind 188, (1934), Paris
[2] Krasnoselski, M.A, Topological methods in the theory of nonlinear integral equations, (1964), Macmillan New York
[3] Schwartz, J.T, Generalizing the Lusternik-schnirelman theory of critical points, Commun. pure appl. math., 17, 307-315, (1964) · Zbl 0152.40801
[4] Palais, R.S, Lusternik-schnirelman theory on Banach manifolds, Topology, 5, 115-132, (1966) · Zbl 0143.35203
[5] Browder, F.E, Infinite dimensional manifolds and nonlinear eigenvalue problems, Ann. of math., 82, 459-477, (1965) · Zbl 0136.12002
[6] Amann, H, Lusternik-schnirelman theory and nonlinear eigenvalue problems, Math. ann., 199, 55-72, (1972)
[7] Clark, D.C, A variant of the Lusternik-schnirelman theory, Indiana univ. math. J., 22, 65-74, (1972) · Zbl 0228.58006
[8] Coffman, C.V, A minimum-maximum principle for a class of nonlinear integral equations, J. analyse math., 22, 391-419, (1969) · Zbl 0179.15601
[9] Coffman, C.V, On a class of nonlinear elliptic boundary value problems, J. math. mech., 19, 351-356, (1970) · Zbl 0194.42103
[10] Hempel, J.A, Superlinear variational boundary value problems and nonuniqueness, ()
[11] Hempel, J.A, Multiple solutions for a class of nonlinear boundary value problems, Indiana univ. math. J., 20, 983-996, (1971) · Zbl 0225.35045
[12] Ambrosetti, A, Esistenza di infinite soluzioni per problemi non lineari in assenza di parametro, Atti accad. naz. lincei mem. cl. sci. fiz. mat. natur. ser. I, 52, 660-667, (1972) · Zbl 0249.35030
[13] \scA. Ambrosetti, On the existence of multiple solutions for a class of nonlinear boundary value problems, Rend. Sem. Mat. Univ. Padova, to appear. · Zbl 0273.35037
[14] \scP. H. Rabinowitz, On pairs of positive solutions for nonlinear elliptic equations, Indiana Univ. Math. J., to appear. · Zbl 0264.35032
[15] \scP. H. Rabinowitz, Variational methods for nonlinear elliptic eigenvalue problems, to appear, Indiana Univ. Math. J. · Zbl 0278.35040
[16] Nehari, Z, On a class of nonlinear integral equations, Math. Z., 72, 175-183, (1959) · Zbl 0092.10903
[17] \scP. H. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain Math. J., to appear. · Zbl 0255.47069
[18] Palais, R.S; Smale, S, A generalized Morse theory, Bull. amer. math. soc., 70, 165-171, (1964) · Zbl 0119.09201
[19] Rabinowitz, P.H, Nonlinear Sturm-Liouville problems for second order ordinary differential equations, Commun. pure appl. math., 23, 939-961, (1970) · Zbl 0206.09706
[20] \scR. E. L. Turner, Superlinear Sturm-Liouville problems, to appear, J. Diff. Eq. · Zbl 0272.34031
[21] Agmon, S, The Lp approach to the Dirichlet problem, Ann. scuolu. norm. sup. Pisa, 13, 405-448, (1959) · Zbl 0093.10601
[22] Berger, M.S; Berger, M.S, A Sturm-Liouville theorem for nonlinear elliptic partial differential equations, Ann. scuola. norm. sup. Pisa, Corrections, 22, 351-354, (1968) · Zbl 0155.16902
[23] Pohozaev, S.I, On the eigenfunctions of quasilinear elliptic problems, Math. USSR-sb., 11, 171-188, (1970) · Zbl 0217.13203
[24] Amann, H, Existence theorems for equations of Hammerstein type, Appl. anal., 1, 385-397, (1972) · Zbl 0244.47047
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