1,720,984 research outputs found
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 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
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
Mountain Pass solutions for quasi-linear equations via a monotonicity trick
No full textWe obtain the existence of symmetric Mountain Pass solutions for quasi-linear equations without the typical assumptions which guarantee the boundedness of an arbitrary Palais– Smale sequence. This is done through a recent version of the monotonicity trick proved in Squassina (in press) [22]. The main results are new also for the p-Laplacian operator
Bounded positive critical points of some multiple integrals of the Calculus of Variations"
No full textMultiple 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
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
Positive solutions for a weakly coupled nonlinear Schrodinger system
No full textExistence of a nontrivial solution is established, via variational methods, for a system of weakly coupled nonlinear Schrodinger equations. The main goal is to obtain a positive solution, of minimal action if possible,
with all vector components not identically zero. Generalizations for nonautonomous systems are considered
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