298 research outputs found
Multiple positive solutions of a scalar field equation in R^n
No full textFrom the viewpoint of the calculus of variations, the perturbed Kazdan-Warner problem:
(1) −∆u+λu=k(x)u^{p−1}, u>0 in R^n, u→0 at ∞,
where n≥3 and p>1 is subcritical. Problem (1) is studied with regard of the effect of the set M on topology of the energy sub levels: in the main results it is shown that the Lyusternik-Schnirelman category of M can
affect the number of positive solutions to (1) in case p is close enough to the critical Sobolev exponent
Ground state solutions of a critical problem involving cylindrical weights
No full textWe prove some existence and non-existence results for a nonlinear elliptic equation involving cylindrical weights and critical growth
ON THE CONTINUITY OF THE NEMITSKY OPERATOR INDUCED BY A LIPSCHITZ CONTINUOUS MAP
No full textLet f be a Lipschitz function of N real variables with values into R^k. Let
Ω be a bounded domain in ,and take .
The Nemitsky operator associated with is defined, for in the Sobolev space
, by . When , is continuous from into . In this paper a counterexample is given to show that the above is not true when . However, additional conditions on are provided which ensure that does map
continuously into . In particular this is so when is Lipschitz with first order derivatives which are continuous outside a closed singular set with empty interior
Many closed K-magnetic geodesics on S^2
No full textIn this paper we adopt an alternative, analytical approach to Arnol’d problem about the existence of closed and embedded K-magnetic geodesics in the round 2-sphere ^2, where :^2→R is a smooth scalar function. In particular, we use Lyapunov-Schmidt finite-dimensional reduction coupled with a local variational formulation in order to get some existence and multiplicity results bypassing the use of symplectic geometric tools such as the celebrated Viterbo’s theorem and Bottkoll results
The role of the spectrum of the laplace operator on S-2 in the H-bubble problem
No full textLet H be a given smooth scalar function on R^3. An H-bubble is an S^2-type parametric surface in R^3 having mean curvature H(p) at every regular point. In the present paper we look for (H(|\cdot|)+\epsilon k)-bubbles, where H
is a radial prescribed curvature satisfying H(1)=1, k is given and
\epsilon is a small parameter
Existence of H-bubbles in a perturbative setting
No full textGiven 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
Entire solutions for a class of variational problems involving the biharmonic operator and Rellich potentials
No full textWe study existence, multiplicity and qualitative properties of entire solutions for a noncompact problem related to second-order interpolation inequalities with weights. More precisely, we deal with the following family of equations
Δu=λ|x|^{-4}u+|x|^{-β}|u|^{q-2}u in R^N,
where N≥5, q>2, β=N-q(N-4)2 and λ∈R is smaller than the Rellich constant
H-bubbles in a perturbative setting: The finite-dimensional reduction method
No full textGiven a regular function H: R^3 → R, we look for H-bubbles, that is, regular surfaces in R^3 parametrized on the sphere S^2, with mean curvature H at every point.
Here we study the case of H(u) = H_0 + εH_1(u) =: H_ε(u), where H_0 is a nonzero constant, ε is the smallness parameter, and H_1 is any C^2-function. We prove that if p̄ ∈ R^3 is a "good" stationary point for
a suitable Melnikov-type function Γ, then for |ε| small there exists an H_ε-bubble ω_ε that converges to a sphere of radius 1/ |H_0| centered at at p̄, as ε → 0
HARDY-SOBOLEV-MAZ'YA INEQUALITIES: SYMMETRY AND BREAKING SYMMETRY OF EXTREMAL FUNCTIONS
No full textDenote points in R^k ×R^{N - k} as pairs ξ = (x,y), and assume 2 ≤ k < N. In this paper, we study the problem
-Δ v=λ|x|^{-2} v+ |x|{-b}v^{p-1} in R^N, x≠ 0, ν > 0
where $p > 2, b = N - pN - 2\2 and λ ≤ (k-2\2)2, the Hardy constant.
We prove existence, symmetry and breaking symmetry results
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