1,720,992 research outputs found
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
A multiplicity result for singularly perturbed problems in topologically nontrivial domains
No full textThe problem -epsilon(2)Deltau + a(epsilon)(x)u = u(p-1) with zero Dirichlet boundary condition is considered in a nontrivial bounded domain Omega subset of R-N. Under the assumption that a, (x) greater than or equal to a(0) > 0 concentrates at a point of Omega as epsilon --> 0 and has a suitable behaviour at infinity and, moreover, that p > 2 and p < 2N/N-2 if N greater than or equal to 3, the existence of at least (catOmega) + 2 distinct positive solutions is proved
Positive bound state solutions for some Schrödinger-Poisson systems
Full text linkThe paper deals with a class of Schrödinger-Poisson systems, where
the coupling term and the other coefficients do not have any symmetry
property. Moreover, the setting we consider does not allow the existence
of ground state solutions. Under suitable assumptions on the decay
rate of the coefficients, we prove existence of a bound state, finite
energy solution
Infinitely many positive standing waves for Schrödinger equations with competing coefficients
No full textThe article deals with the equation Delta u+ a(x)u + b(x)u(q) - u(p) = 0 u is an element of H-1 (R-N), with N > 2,1 < q 3, infa> 0 a(x) -> a(infinity) and b(x) -> 0 as vertical bar x vertical bar -> infinity When (2(X) < a. and b(x) 0, only a finite number of positive solutions to the problem is reasonably. expected. Here we prove that the, presence of a nonzero term b(x)Liq with b(x) > 0, b(x) 4 0, under suitable assumptions on the decay rates of a and b, allows to obtain infinitely many posiive solutions
The effect of concentrating potentials in some singularly perturbed problems
No full textThe equation -epsilon(2) Deltau + a(epsilon)(x)u = f(u) with boundary Dirichlet zero data is considered in a bounded domain Omega subset of R-N. Under the assumption that a(epsilon)(x) greater than or equal to a(infinity) > 0 concentrates, as epsilon --> 0, round a manifold M is an element of Omega and that f is a superlinear function, satisfying suitable growth assumptions, the existence of multiple distinct positive solutions is proved
On Some Scalar Field Equations with Competing Coefficients
No full textThis paper deals with semilinear elliptic problems of the type −Δu + α(x)u = β(x)|u|p−1u in RN, u(x) >0 inRN, u ∈ H1(RN), where p is superlinear but subcritical and the coefficients α and β are positive functions such that α(x) → a∞ > 0 and β(x) → b∞ > 0, as |x| → ∞. Aim of this work is to describe some phenomena that can occur when the coefficients are “competing.
Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology
No full textWe use Morse theory to estimate the number of positive solutions of an elliptic problem in an open bounded set OMEGA subset-of R(N). The number of solutions depends on the topology of OMEGA, actually on P(t)(OMEGA), the Poincare polynomial of OMEGA. More precisely, we obtain the following Morse relations: SIGMA(u is-an-element-of K) t(mu(u)) = tP(t)(OMEGA) + t2 [P(t)(OMEGA) - 1] + t(1 + t) Q(t), where Q(t) is a polynomial with non-negative integer coefficients, K is the set of positive solutions of our problem and mu(u) is the Morse index of the solution u
Positive solutions for some Schrodinger equations having partially periodic potentials
No full textAbstractThis paper is concerned with the problem of finding positive solutions u∈H01(Ω) of the equation −Δu+(a∞+a(x))u=|u|q−2u, where q is subcritical, Ω is either RN or an unbounded domain which is periodic in the first p coordinates and whose complement is contained in a cylinder {(x′,x″)∈Rp×RN−p:|x″|<R}, a∞>0, a∈C(RN,R) is periodic in the first p coordinates, infx∈RN(a∞+a(x))>0 and a(x′,x″)→0 as |x″|→∞ uniformly in x′. The cases a⩽0 and a⩾0 are considered and it is shown that, under appropriate assumptions on a, the problem has one solution in the first case and p+1 solutions in the second case when p⩽N−2
Multiple positive bound states for critical Schrödinger-Poisson systems
No full textUsing variational methods we prove some results about existence and multiplicity of positive bound states of to the following Schrodinger-Poisson system (SP): -Delta u+V(x)u+K(x)phi(x)u=u^5; -Delta phi =K(x)u^2 x in R^3. We remark that (SP) exhibits a ``double'' lack of compactness because of the unboundedness of R^3 and the critical growth of the nonlinear term and that in our assumptions ground state solutions of (SP) do not exist.
"The research that led to the present paper was partially supported by a grant of the group GNAMPA of INdAM
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