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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

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: 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, (thesis (1970), University of New England: University of New England Australia)
[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
[16] Nehari, Z., On a class of nonlinear integral equations, Math. Z., 72, 175-183 (1959) · Zbl 0092.10903
[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
[21] Agmon, S., The \(L_p\) approach to the Dirichlet problem, Ann. Scuolu. Norm. Sup. Pisa, 13, 405-448 (1959) · Zbl 0093.10601
[22] Berger, M. S., 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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