1,720,986 research outputs found
Multiple homoclinic solutions for a class of autonomous singular systems in R2.
No full textWe look for homoclinic solutions for a class of second order autonomous Hamiltonian systems in R-2 with a potential V having a strict global maximum at the origin and a finite set S subset of R-2 of singularities, namely V(x) --> -infinity as dist(x,S) --> 0. We prove that if V satisfies a suitable geometrical property then for any k epsilon N the system admits a homoclinic orbit turning k times around a singularity xi epsilon S
Existence and non existence results for a class of nonlinear singular Sturm-Liouville equations
No full textWe study the existence and nonexistence of solutions for a class of equations of the form
−(ωu′)′ = |u|^{p−2}u in R
where p > 2 and ω is a nonnegative continuous function with isolated zeroes of power type
Existence of H-bubbles in a perturbative setting
Full text linkGiven a C^1 function H: R^3 → R, we look for H-bubbles, i.e., surfaces in R^3 parametrized by the sphere S^2 with mean curvature H at every regular point. Here we study the case H(u)=H_0(u)+∈H_1(u) where H_0 is some "good" curvature (for which there exist H_ 0-bubbles with minimal energy, uniformly bounded in L^∞), ∈ is the smallness parameter, and H_1 is any C^1 functio
On a variational degenerate elliptic problem
No full textIn this paper we study a class of variational degenerate elliptic problems of the form
-div(a(x) Δu) = f(x,u) in Ω,
u = 0 on ∂Ω,
where Ω is a bounded or unbounded domain in R^n, n ≥ 2
On the existence of extremal functions for a weighted Sobolev embedding with critical exponent
No full textWe investigate the existence of ground state solutions to the Dirichlet problem
-div(|x|α∇u) = |u|2*α-2u in Ω,
u = 0 on ∂Ω,
where α ε (0,2), 2*α = 2n/n-2+α and Ω is a domain in R^n. In particular we prove that a non negative ground state solution exists when the domain Ω is a cone, including the case Ω = R^n. Moroever, we study the case of arbitrary domains, showing how the geometry of the domain near the origin and at infinity affects the existence or non existence of ground state solutions
Existence of minimal H-bubbles
No full textGiven a function H ∈ C1(R3) asymptotic to a constant at infinity, we investigate the existence of H-bubbles, i.e., nontrivial, conformal surfaces parametrized by the sphere, with mean curvature H. Under some global hypotheses we prove the existence of H-bubbles with minimal energy
The Dirichlet Problem for H -Systems with Small Boundary Data: BlowUp Phenomena and Nonexistence Results
No full textGiven H: R^3 → R of class C^1 and bounded, we consider a sequence (u_n) of solutions of the H-system
Δu = 2H(u)u_x ∧ u_y
in the unit open disc D satisfying the boundary condition u_n = y_n on ∂D. In the first part of this paper, assuming that (u_n) is bounded in H^1 (D, R^3) we study the behavior of (u_n) when the boundary data γ_n shrink to zero. We show that either u_n → 0 strongly in H^1 (D, R^3) or u_n blows up at least one H-bubble ω, namely a nonconstant, conformal solution of the H-system on R^2. Under additional assumptions on H, we can obtain more precise information on the blow up. In the second part of this paper we investigate the multiplicity of solutions for the Dirichlet problem on the disc with small boundary datum. We detect a family of nonconstant functions H (even close to a nonzero constant in any reasonable topology) for which the Dirichlet problem cannot admit a "large" solution at a mountain pass level when the boundary datum is small
On a class of two-dimensional singular elliptic problems
No full textWe consider Dirichlet problems of the form
-|x|^α Δu = λu + g(u) in Ω,
u = 0 on ∂Ω,
where α, λ ∈ R, g ∈ C(R) is a superlinear and subcritical function, and Ω is a domain in R^2. We study the existence of positive solutions with respect to the values of the parameters α and λ, and according that 0 ∈ Ω or 0 ∈ ∂Ω, and that Ω is an exterior domain or not
Measure properties of the set of initial data yielding nonuniqueness for a class of differential inclusions.
Full text link- …
