1,720,973 research outputs found
On the multiplicity of positive solutions for p-Laplace equations via Morse theory
No full textLet us consider the quasilinear problem(P_ε)
{(-ε^p Δ_p u + u^{p-1} =f(u), in Ω,;
u>0, in Ω,;
u=0, on ∂Ω) where Ω is a bounded domain in R^N with smooth boundary, N > p, 2 ≤ p 0 is a parameter. We prove that there exists ε* > 0 such that, for any ε ∈]0,ε*[, (P_ε) has at least 2P_1(Ω)-1 solutions, possibly counted with their multiplicities, where P_t(Ω) is the Poincaré polynomial of Ω. Using Morse techniques, we furnish an interpretation of the multiplicity of a solution, in terms of positive distinct solutions of a quasilinear equation on Ω, approximating (P_ε)
Critical groups computations on a class of Sobolev Banach spaces via Morse index
No full textIn 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|^2dx+∫_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_0^{1,p}(Ω) to its dual space, so that the generalized Morse splitting lemma does not work. In spite of the lack of a Hilbert structure, we compute the critical groups of f in u via its Morse index
Morse index and critical groups for p-Laplace equations with critical exponents
No full textIn this work we consider a class of Euler functionals defined in Banach spaces, associated to quasilinear elliptic problems involving the critical Sobolev exponent. We perform critical groups estimates via the Morse index
Multiple positive solutions for a critical quasilinear equation via Morse theory
No full textWe deal with the existence of solutions for the quasilinear problem(P_λ) {(- Δ_p u = λ u^{q-1} + u^{p*-1}, in Ω,;
u > 0, in Ω,;
u = 0, on ∂ Ω,)
where Ω is a bounded domain in R^N with smooth boundary, N≥p^2, 10 is a parameter. Using Morse techniques in a Banach setting, we prove that there exists λ* > 0 such that, for any λ ∈ (0, λ*), (P_λ) has at least P_1(Ω) solutions, possibly counted with their multiplicities, where P_t (Ω) is the Poincaré polynomial of Ω. Moreover for p ≥ 2 we prove that, for each λ ∈ (0, λ*), there exists a sequence of quasilinear problems, approximating (P_λ), each of them having at least P_1(Ω) distinct positive solutions
Marino-Prodi perturbation type results and Morse indices of minimax critical points for a class of functionals in Banach spaces
No full textIn this work, we study a class of Euler functionals defined in Banach spaces, associated with quasilinear elliptic problems involving p-Laplace operator (p>2). First we obtain perturbation results in the spirit of the remarkable paper by Marino and Prodi (Boll. U.M.I. (4) 11(Suppl. fasc. 3): 1-32, 1975), using the new definition of nondegeneracy given in (Cingolani- Vannella, Ann. Inst. H. Poincaré: Analyse Non Linéaire. 20:271-292, 2003). We also extend Morse index estimates for minimax critical points, introduced by Lazer and Solimini (Nonlinear Anal. T.M.A. 12:761-775, 1988) in the Hilbert case, to our Banach setting
Some results on critical groups for a class of functionals defined on Sobolev Banach spaces
No full textWe present
critical groups estimates for a functional f defined on the
Banach space , where Ω is a bounded domain in
R^N, p>2, associated to a quasilinear elliptic
equation involving p-laplacian. In spite of the lack of an
Hilbert structure and of Fredholm property of the second
order differential of f in each critical point, we compute the critical
groups of f in each isolated critical point via Morse index
Morse index computations for a class of functionals defined in Banach spaces
No full textIn this paper we focus our attention on the estimates of critical groups for some functionals associated to a class of quasilinear equations, involving p-Laplacian
Multiplicity results for a quasilinear elliptic system via Morse theory
No full textIn this work we prove some multiplicity results for solutions of a system of elliptic quasilinear equations, involving the p-Laplace operator (p > 2). The proofs are based
on variational and topological arguments and make use of new perturbation results in Morse theory for the Banach space W^{1,p}_0
Multiplicity results for a quasilinear elliptic system via Morse theory
No full textIn this work we prove some multiplicity results for solutions of a system of elliptic quasilinear equations, involving the p-Laplace operator (p > 2). The proofs are based on variational and topological arguments and make use of new perturbation results in Morse theory for the Banach space W_0^{1,p}
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