1,720,994 research outputs found
Critical Points for Non Differentiable Functionals
No full textIn this paper we deal with the existence and multiplicity of critical points for non differentiable integral functionals defined in the Sobolev space W1,p(Ω) (p > 1) by:
0
where Ω is a bounded open set of RN, with N ≥ 3 and p ≤ N. Under natural assump- tions F turns out to be not Frech ́et differentiable on W1,p(Ω), thus classical critical
point theory cannot be applied. The existence of a critical point of F has been proved in [1] by means of a suitable extension of the Ambrosetti-Rabinowitz minimax result. Here we get existence and multiplicity of critical points of F applying a generalization of a symmetric version of the Mountain-Pass theorem proved in [10]. We will follow the same procedure of [7] where the quasilinear case has been treated
Quasi--Linear equations on R^N: Perturbation Results
No full textIn this paper we prove existence of nontrivial solutions for the quasi-linear elliptic problem
{div((I + epsilonA(x, u))delu) + u + epsilonH(x, u,delu) = \u\(p-1), in R-N, u is an element of H-1(R-N) boolean AND W-2,W-q (R-N), q > N
where 1 2 and the operator -div((I + epsilonA(x, u))delu) +epsilonH(x, u, delu) is a perturbation of the Laplacian. We use a perturbation method recently developed in [1], [2], [3] and we get results both in the variational and in the non-variational framework
Critical Points for Some Functionals of the Calculus of Variations
No full textIn this paper we prove the existence of critical
points of non differentiable functionals of the kind
J(v)=\frac_\Omega A(x,v)\nabla v\cdot\nabla v-\frac1{p+1}\int_\Omega
(v^+)^{p+1},
where , if and
stands for the positive part of the function . The
coefficient is a Carathéodory matrix
derivable with respect to the variable . Even if both
and are uniformly bounded
by positive constants, the functional fails
to be differentiable on . Indeed, is only
derivable along directions of
so that the classical critical point theory cannot be
applied.
We will prove the existence of a critical point of by
assuming that there exist positive continuous functions
, and a positive constants
and satisfying , ,
, with in
Positive solutions for asymptotically linear problems in exterior domains
No full textThe existence of a positive solution for a class of asymptotically linear problems in exterior domains is established via a linking argument on the Nehari manifold and by means of a barycenter function
Multiple critical points for nondifferentiable functionals involving Hardy potentials
No full textIn this paper we study general functionals of the calculus of variations with the presence of a Hardy potential. We will improve several results obtained in the semilinear framework. We will first prove a general weak lower
semicontinuity result, which will imply the existence of a minimum point whenever the functional is coercive. Then we will demonstrate existence and multiplicity results of critical points, even if our functional is not differentiable. We will apply a nonsmooth critical point theory developed in Corvellec et al. (Nonlinear Anal. 1 (1993) 151) and Degiovanni and Marzocchi (Ann. Mat. Pura Appl. 167 (1994) 73)
Orbital Stability of ground state solutions of coupled nonlinear Schr"odinger equations
No full textOrbital stability property for weakly coupled nonlinear
Schrodinger equations is investigated. Different families of orbitally stable standing waves solutions will be found, generated by different classes of solutions of the associated elliptic problem. In particular, orbitally stable standing waves can be generated by least action solutions, but also by solutions with one trivial component whether or not they are ground states. Moreover, standing waves with components propagating with the same frequencies are orbitally stable if generated by vector solutions of a suitable Schro ̈dinger weakly coupled system, even if they are not ground states
Singularly perturbed elliptic problems with nonautonomous asymptotically linear nonlinearities
No full textWe consider a class of singularly perturbed elliptic problems with nonautonomous asymptotically linear nonlinearities. The dependence on the spatial coordinates comes from the presence of a potential and of a function representing a saturation effect. We investigate the existence of nontrivial nonnegative solutions concentrating around local minima of both the potential and of the saturation function. Necessary conditions to locate the possible concentration points are also given
An upper bound for the least energy of a sign-changing solution to a zero mass problem
No full textWe give an upper bound for the least possible energy of a sign-changing solution to the nonlinear scalar field equation -Delta u = f(u), u is an element of D-1,D-2(R-N), where N >= 5 and the nonlinearity f is subcritical at infinity and supercritical near the origin. More precisely, we establish the existence of a nonradial sign-changing solution whose energy is smaller that 12c(0) if N = 5, 6 and smaller than 10c(0) if N >= 7, where c(0) is the ground state energy
Weakly coupled nonlinear Schrodinger systems: the saturation effect
No full textWe study the existence of solutions for a class of saturable weakly coupled Schrödinger systems. In most of the cases we show that least energy solutions have neces- sarily one trivial component. In addition sufficient conditions for the existence of a solution with both positive components are found
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