## 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

### MSC:

 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces 35J20 Variational methods for second-order elliptic equations
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### References:

 [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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