# zbMATH — the first resource for mathematics

Connecting orbits for nonlinear differential equations at resonance. (English) Zbl 1302.34098
The author examinates the problem of the existence of orbits connecting stationary points of the following equation $u'(t)=-Au(t)+\lambda u(t)+F(u(t))\quad\text{for }\;t>0,$ where $$\lambda$$ is an eigenvalue of a sectorial operator $$A: X\supset D(A) \to X$$ defined on a real Banach space $$X$$ and $$F: X^{\alpha} \to X$$ is a continuous mapping. $$X^{\alpha}$$ for $$\alpha \in (0,1)$$ denotes here a fractional space defined by the equality $$X^{\alpha}=D((A+\delta I)^{\alpha})$$, where $$\delta >0$$ is chosen in such a way that $$A+\delta I$$ is a positive definite operator. It is assumed that the above equation is at resonance at infinity, that is ker$$(\lambda I-A)\neq \{0\}$$ and $$F$$ is a bounded mapping.
For that purpose, imposing special geometrical conditions on the nonlinearity, the author proves the index formula for bounded orbits which is the tool to determine the Conley index for the maximal invariant set contained in a sufficiently large ball.
It is established that those geometrical conditions encompass the well-known Landesman- Lazer conditions as well as the strong resonance ones.
As an example the author considers the existence of orbits connecting stationary points for the heat equation being at resonance at infinity.

##### MSC:
 34G20 Nonlinear differential equations in abstract spaces 34C37 Homoclinic and heteroclinic solutions to ordinary differential equations 37B30 Index theory for dynamical systems, Morse-Conley indices
##### Keywords:
Conley index; connecting orbit; resonance; semiflow; semigroup.
Full Text:
##### References:
 [1] Ambrosetti, Antonio; Mancini, Giovanni, Theorems of existence and multiplicity for nonlinear elliptic problems with noninvertible linear part, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 5, 1, 15-28, (1978) · Zbl 0375.35024 [2] Bartolo, P.; Benci, V.; Fortunato, D., Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity, Nonlinear Anal., 7, 9, 981-1012, (1983) · Zbl 0522.58012 [3] Brézis, H.; Nirenberg, L., Characterizations of the ranges of some nonlinear operators and applications to boundary value problems, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 5, 2, 225-326, (1978) · Zbl 0386.47035 [4] Conley, Charles, Isolated invariant sets and the Morse index, CBMS Reg. Conf. Ser. Math., vol. 38, (1978), American Mathematical Society Providence, RI · Zbl 0397.34056 [5] Cholewa, Jan W.; Dlotko, Tomasz, Global attractors in abstract parabolic problems, London Math. Soc. Lecture Note Ser., vol. 278, (2000), Cambridge University Press Cambridge · Zbl 0954.35002 [6] de Figueiredo, Djairo G.; Gossez, Jean-Pierre, Strict monotonicity of eigenvalues and unique continuation, Comm. Partial Differential Equations, 17, 1-2, 339-346, (1992) · Zbl 0777.35042 [7] Garofalo, Nicola; Lin, Fang-Hua, Unique continuation for elliptic operators: A geometric-variational approach, Comm. Pure Appl. Math., 40, 3, 347-366, (1987) · Zbl 0674.35007 [8] Henry, Daniel, Geometric theory of semilinear parabolic equations, Lecture Notes in Math., vol. 840, (1981), Springer-Verlag Berlin · Zbl 0456.35001 [9] Kokocki, Piotr, Averaging principle and periodic solutions for nonlinear evolution equations at resonance, Nonlinear Anal., 85, 253-278, (2013) · Zbl 1292.34059 [10] Landesman, E. M.; Lazer, A. C., Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech., 19, 609-623, (1969/1970) · Zbl 0193.39203 [11] Prizzi, Martino, On admissibility for parabolic equations in $$\mathbb{R}^n$$, Fund. Math., 176, 3, 261-275, (2003) · Zbl 1022.37013 [12] Rybakowski, Krzysztof P., Trajectories joining critical points of nonlinear parabolic and hyperbolic partial differential equations, J. Differential Equations, 51, 2, 182-212, (1984) · Zbl 0529.35040 [13] Rybakowski, Krzysztof P., On the homotopy index for infinite-dimensional semiflows, Trans. Amer. Math. Soc., 269, 2, 351-382, (1982) · Zbl 0468.58016 [14] Rybakowski, Krzysztof P., Nontrivial solutions of elliptic boundary value problems with resonance at zero, Ann. Mat. Pura Appl. (4), 139, 237-277, (1985) · Zbl 0572.35037 [15] Rybakowski, Krzysztof P., The homotopy index and partial differential equations, Universitext, (1987), Springer-Verlag Berlin · Zbl 0628.58006 [16] Salamon, Dietmar, Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc., 291, 1, 1-41, (1985) · Zbl 0573.58020 [17] Triebel, H., Interpolation theory, function spaces, differential operators, (1978), VEB Deutscher Verlag der Wissenschaften Berlin · Zbl 0387.46033 [18] Valdo, José; Gonçalves, A., On bounded nonlinear perturbations of an elliptic equation at resonance, Nonlinear Anal., 5, 1, 57-60, (1981) · Zbl 0473.35040
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.