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 $2{mathcal P}-1(M) -1$ nonconstant solutions u H1,p(M) to the quasilinear problem $$(P-epsilon) left{{egin{array}{@{}l@{}} -epsilon^p Delta-{p,g} u +u^{p-1}=u^{q-1}, \u>0,end{array}} ight$$ for ε > 0 small enough, where 2 ≤ p < n, p < q < p∗, p∗= np/(n - p) and $Delta-{p, g} u = extrm{div}-g (ert abla uert-{g}^{p-2} abla u)$ is the p-laplacian associated to g of u (note that Δ2,g = Δg) and ${mathcal P}-t(M)$ denotes the Poincaré polynomial of M. We also establish results of genericity of nondegenerate solutions for the quasilinear elliptic problem (Pε).

Cingolani, S., Vannella, G., Visetti, D. (2015). Multiplicity and nondegeneracy of positive solutions to quasilinear equations on compact Riemannian manifolds. COMMUNICATIONS IN CONTEMPORARY MATHEMATICS, 17(2) [10.1142/S0219199714500291].

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

Visetti D
2015

Abstract

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 $2{mathcal P}-1(M) -1$ nonconstant solutions u H1,p(M) to the quasilinear problem $$(P-epsilon) left{{egin{array}{@{}l@{}} -epsilon^p Delta-{p,g} u +u^{p-1}=u^{q-1}, \u>0,end{array}} ight$$ for ε > 0 small enough, where 2 ≤ p < n, p < q < p∗, p∗= np/(n - p) and $Delta-{p, g} u = extrm{div}-g (ert abla uert-{g}^{p-2} abla u)$ is the p-laplacian associated to g of u (note that Δ2,g = Δg) and ${mathcal P}-t(M)$ denotes the Poincaré polynomial of M. We also establish results of genericity of nondegenerate solutions for the quasilinear elliptic problem (Pε).
Articolo in rivista - Articolo scientifico
Morse index; perturbation results; positive solutions; Quasilinear elliptic equations; Riemannian manifold;
English
41
Cingolani, S., Vannella, G., Visetti, D. (2015). Multiplicity and nondegeneracy of positive solutions to quasilinear equations on compact Riemannian manifolds. COMMUNICATIONS IN CONTEMPORARY MATHEMATICS, 17(2) [10.1142/S0219199714500291].
Cingolani, S; Vannella, G; Visetti, D
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/355731
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