Giuseppina Vannella

Critical groups computations on a class of Sobolev Banach spaces via Morse index

Abstract In this paper we deal with critical groups estimates for a functional f :W 0 1,p ( Ω )→ R (p>2), Ω bounded domain of R N , defined by setting f(u)= 1 p ∫ Ω |∇u| p dx+ 1 2 ∫ Ω |∇u| 2 dx+ ∫ Ω G(u) dx where G(t)=∫ 0 t g(s) ds and g is a smooth real function on R , growing subcritically. We remark that the second derivative of f in each critical point u is not a Fredholm operator from W 1,p 0 ( Ω ) to its dual space, so that the generalized Morse splitting lemma does not work. In spite of the lack of an Hilbert structure, we compute the critical groups of f in u via its Morse index.

MULTIPLICITY RESULTS FOR A QUASILINEAR ELLIPTIC SYSTEM VIA MORSE THEORY

In this work we prove some multiplicity results for solutions of a system of elliptic quasilinear equations, involving the p-Laplace operator (p > 2). The proof are based on variational and topological arguments and makes use of new perturbation results in Morse theory for the Banach space

Morse Index Computations for a Class of Functionals Defined in Banach Spaces

W^{1,p}_0

Multiplicity and nondegeneracy of positive solutions to quasilinear equations on compact Riemannian manifolds

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Regularity and Morse Index of the Solutions to Critical Quasilinear Elliptic Systems

In Morse theory the behavior of a C2Euler functionalF, defined on a Hilbert spaceH,near its critical points can be described by the estimates of thecritical groupsin the critical points. For convenience of the reader we recall the definition of critical group. For any a e R, we denote. Moreover let u be a critical point ofF, at levelc = F(u).We call the qth critical group ofFat u,q =0, 1, 2,…, whereHq (A,B)stands for the qth Alexander-Spanier cohomology group of the pair (A,B)with coefficients in 1K (cf. [2]).

Some qualitative properties of the solutions of an elliptic equation via Morse theory

We consider a compact, connected, orientable, boundaryless Riemannian manifold (M, g) of class C∞ where g denotes the metric tensor. Let n = dim M ≥ 3. Using Morse techniques, we prove the existence of nonconstant solutions u ∈ H1,p(M) to the quasilinear problem for e > 0 small enough, where 2 ≤ p < n, p < q < p*, p* = np/(n - p) and is the p-laplacian associated to g of u (note that Δ2,g = Δg) and denotes the Poincare polynomial of M. We also establish results of genericity of nondegenerate solutions for the quasilinear elliptic problem (Pe).

Amann–Zehnder type results for p-Laplace problems

Regularity results and critical group estimates are studied for critical (p, r)-systems. Multiplicity results of solutions for a critical potential quasilinear system are also proved using Morse theory.

Critical Group Estimates for Nonregular Critical Points of Functionals Associated With Quasilinear Elliptic Equations

where Ω ⊂ R is an open bounded domain with sufficiently regular boundary (n ≥ 3), ε > 0 is a real number and F ∈ C(R) is a real function which satisfies the following assumptions: (i) F is even, (ii) 0 is a local maximum for F , with F (0) = 1 and F ′′(0) < 0, (iii) F (R) ⊂ R and F vanishes at (and only at) 1 and −1, (iv) ∃a > 0 ∀t ≥ 1, F ′′(t) ≥ a, (v) ∃p ∈ ]2, 2∗[ ∃b, c ≥ 0 ∀t ∈ R, |F ′′(t)| ≤ b|t|p−2+c (here 2∗ = 2n/(n− 2)). Let us remark that from (iii) it follows that F ′(1) = F ′(−1) = 0. Moreover, since F ′′ is even and continuous, we see from (ii) and (iii) that ∃β ∈ ]0, 1[ ∀t ∈ ]−β, β[, F ′′(t) < 0 and F ′′(−β) = F ′′(β) = 0.

Morse theory applied to a

The existence of a nontrivial solution is proved for a class of quasilinear elliptic equations involving, as principal part, either the p-Laplace operator or the operator related to the p-area functional, and a nonlinearity with p-linear growth at infinity. To this aim, Morse theory techniques are combined with critical groups estimates.

T^{2}

We consider a class of quasilinear elliptic equations whose principal part includes the p-area and the p-Laplace operators, when p lies in a suitable left neighborhood of 2. For the critical points of the associated functional, we provide estimates of the corresponding critical groups, under assumptions that do not guarantee any further regularity of the critical point.