1,721,059 research outputs found
Variational properties of the first curve of the Fučík spectrum for elliptic operators
Full text linkIn this paper we present a new variational characteriztion of the
first nontrival curve of the Fucik spectrum for elliptic
operators with Dirichlet boundary conditions.
Moreover, we describe the asymptotic behaviour and some properties of
this curve and of the corresponding eigenfunctions.
In particular, this new characterization allows us to compare the
first curve of the Fucik spectrum with the infinitely many
curves we obtained in previous works; for example,
we show that these curves are all asymptotic to the same lines as the
first curve, but they are all distinct from such a curve
Semilinear elliptic problems in unbounded domains with unbounded boundary
No full textThis paper deals with a class of singularly perturbed nonlinear elliptic problems (P-epsilon) with subcritical nonlinearity. The coefficient of the linear part is assumed to concentrate in a point of the domain, as epsilon --> 0, and the domain is supposed to be unbounded and with unbounded boundary. Domains that enlarge at infinity, and whose boundary flattens or shrinks at infinity, are considered. It is proved that in such domains problem (P-epsilon) has at least 2 solutions
Normalized positive solutions for Schrödinger equations with potentials in unbounded domains
Full text linkThe paper deals with the existence of positive solutions with prescribed norm for the Schrodinger equation where or is a compact set, , (also is allowed), . The existence of a positive solution is proved when verifies a suitable decay assumption (D?), or if is small, for some ( if ). No smallness assumption on is required if the decay assumption (D?) is fulfilled. There are no assumptions on the size of . The solution is a bound state and no ground state solution exists, up to the autonomous case and
Positive solutions for autonomous and non-autonomous nonlinear critical elliptic problems in unbounded domains
Full text linkThe paper concerns with positive solutions of problems of the type -Δu+a(x)u=up-1+εu2∗-1 in Ω ⊆ RN, N≥ 3 , 2∗=2NN-2, 2 0. First, some existence results of ground state solutions are proved. Then the case a(x) ≥ a∞ is considered, with a(x) ≢ a∞ or Ω ≠ RN. In such a case, no ground state solution exists and the existence of a bound state solution is proved, for small ε
Positive solutions for a nonlinear elliptic problem with strong lack of compactness
No full textThis paper deals with the lack of compactness in the nonlinear elliptic problem -Delta u+u = vertical bar u vertical bar(p-2)u in Omega, u > 0 in Omega, u = 0 on delta Omega, when Omega is un unbounded domain in R-n and 2 < p < 2n/(n - 2). In particular, a domain Omega is provided where non-converging Palais-Smale sequences exist at every energy level. Nevertheless, it is proved that the problem has infinitely many solutions on Omega
Normalized Schrödinger equations with mass-supercritical nonlinearity in exterior domains
No full textWe consider the problem m −∆u + λu = |u|p−p−2u, where u ∈ H10 (Ω) satisfies |u|2 = m > 0, λ ∈ R and Ω is a smooth exterior domain. We prove the existence of a positive solution with a constrained Morse index less or equal than N + 1 and λ ≥ 0. We treat both the cases m fixed and RN \ Ω small and Ω fixed and m large
On a planar Schrödinger-Poisson system involving a non-symmetric potential
No full textWe prove the existence of a ground state positive solution of Schrödinger-Poisson systems in the plane of the form where p≥4, γ, b>0 and the potential V is assumed to be positive and unbounded at infinity. On the potential we do not require any symmetry or periodicity assumption, and it is not supposed it has a limit at infinity. We approach the problem by variational methods, using a variant of the mountain pass theorem and the Cerami compactness condition
Multiple positive solutions for nonautonomous quasicritical elliptic problems in unbounded domains
No full textThe problem -Delta u + a(epsilon)(x)u = u(N+2/N-2-epsilon), epsilon > 0, with boundary Dirichlet zero data is considered in an exterior domain Omega subset of R-N Assuming that, as epsilon -> 0, a(epsilon) concentrates and blows up at a point of Omega, namely a(epsilon)(x) = a(0) + 1/(epsilon)2aa (x-x(0/)epsilon(a))alpha is an element of R+\{0}, x(0) is an element of Omega, the existence of at least 2 distinct positive solutions is proved, if vertical bar a vertical bar(LN/2) is suitably small. Furthermore, if a(epsilon) (x) has a Suitable behaviour at infinity, the existence of another positive solution is shown
Infinitely many solutions for elliptic equations with non-symmetric nonlinearities
No full textWe deal with the existence of infinitely many solutions for a class of elliptic problems with non-symmetric nonlinearities. Our result, which is motivated by a well known conjecture formulated by A. Bahri and P.L. Lions, suggests a new approach to tackle these problems. The proof is based on a method which does not require to use techniques of deformation from the symmetry and may be applied to more general non-symmetric problems
Nonexistence of solutions for elliptic equations with supercritical nonlinearity in nearly nontrivial domains
No full textWe deal with nonlinear elliptic Dirichlet problems of the form
where is a bounded domain in , , p> 1 and
has supercritical growth from the viewpoint of Sobolev embedding.
o Our aim is to show that there exist bounded contractible non
star-shaped domains , arbitrarily close to domains with
nontrivial topology, such that the problem does not have nontrivial
solutions.
For example, we prove that if , 1<2, with
q>{2pover 2-p} and Omega={(
hocos heta,
hosin heta) :
| heta|{2pover 2-p} there exists ar s>0 such that the
problem has only the trivial solution for all and
- …
