117,392 research outputs found
Existence of solutions for gradient coupled Dirichlet systems
No full textIn this paper, we prove existence of weak solutions in W1,20 (Ω) ∩L∞(Ω) for the gradient coupled Dirichlet system
{u ∈ W1,20 (Ω): –div(M(x)∇u) + u + a(x)∇u · ∇ψ = f(x),
{ψ ∈ W1,20 (Ω): –div(M(x)∇ψ) + ψ + a(x)∇u · ∇ψ = g(x).
We also prove that if f (x), g(x) ≥ 0 (of course ≢ 0 a.e.), then u(x), ψ(x) ≥ 0 and the sets {u = 0} and {ψ = 0} have zero Lebesgue measure
Harnack type inequalities and quantization for the Uniformly Elliptic Liouville Equation
No full textWe extend the Harnack type inequality proved in C. R. Acad. Sci. Paris 315(2) (1992), 159-164, to the solutions of -div(A del u) = Ve(u) in Omega, with no boundary conditions. Here A is a symmetric, uniformly elliptic matrix and Omega subset of R-2 is open and bounded. As an application we are able to generalize the quantization results of Ind. Univ. Math. J. 43(4) (1994), 1255-1270, to the uniformly elliptic case
Uniformly Elliptic Liouville Type Equations: Concentration Compactness and a Priori Estimates
No full textWe analyze the singular behavior of the Green's function for uniformly elliptic equations on smooth and bounded two dimensional domains. Then, we are able to generalize to the uniformly elliptic case some sharp estimates for Liouville type equations due to Brezis-Merle [7] and, in the same spirit of [3], a "mass" quantization result due to Y.Y. Li [21]. As a consequence, we obtain uniform a priori estimates for solutions of the corresponding Dirichlet problem. Then, we improve the standard existence theorem derived by direct minimization and, in the same spirit of [17] and [37], obtain the existence of Mountain Pass type solutions
Existence of finite energy solutions for elliptic systems with L-1-valued nonlinearities
No full textIn this paper, we are going to study the following elliptic system: {-div (b(x, z)del u) = f(x) in Omega, -div(a(x, z)del z) = b(x, z)vertical bar del u vertical bar(2) in Omega, u = 0, z = 0 on partial derivative Omega, where Omega is a bounded open subset of R-N, a(x, s) and b(x, s) are positive and coercive Caratheodory functions, and f is an element of L-m(Omega). The main purpose of this paper is to prove existence and regularity results with an improved regularity of the function z in the class of Sobolev spaces, and the existence of solutions (u, z) both with finite energy
Very singular solutions for linear Dirichlet problems with singular convection terms
No full textWe study the existence of distributional solutions for the boundary value problems (1.1) and (1.2) if E does not belong to LN, namely [Formula presented], A∈R. The size of A plays an important role: if α(N−2)≤|A
A semilinear system of Schrödinger–Maxwell equations
No full textIn this paper we are going to prove existence and regularity results for positive solutions of the following elliptic system: −div(M(x)∇u)+rφur−1=f+φr,−div(M(x)∇φ)+ruφr−1=ur.where Ω is a bounded open subset of RN, M is a bounded, uniformly elliptic matrix, r>1, and f≥0 belongs to some Lebesgue space Lm(Ω), with m≥1. We will also prove the relationships of the solutions of the system with saddle points of the integral functional [Formula presented
Uniform Estimates and Blow-up Analysis for the Emden Exponential Equation in Any Dimension
No full textWe study the asymptotic behavior as n ->infinity of a sequence of functions u(n) satisfying the Emden equation Delta u(n) = e(un) in a bounded domain Omega subset of R-N, with N >= 2. By assuming a suitable uniform bound and an additional monotonicity property, we prove that the *-weak limit in the sense of measures of a subsequence of e(un) is either a function of L-1(Omega), or a purely singular measure concentrated on an (N - 2)-rectifiable set
A consequence of Djairo’s Lectures on the Ekeland variational principle
No full textIn this paper we prove that if a functional has bounded minimum u, then is is possible, using Ekeland’s ε-variational principle, to build a minimizing sequence which is uniformly convergent to u
Existence Results for a System of Kirchhoff–Schrödinger–Maxwell Equations
No full textIn this paper, we study existence, nonexistence, and properties of solutions for some Kirchhoff–Schrödinger–Maxwell systems as (1.3). The solutions can be seen as saddle points of functionals which are unbounded both from above and from below
A weak minima approach to the study of the existence of saddle points of integral functionals
No full textWe study of the existence of saddle points of the functional [Formula presented] defined in (1.1) both in the regular case, i.e., if [Formula presented] belongs to [Formula presented], and in the singular one, i.e., if [Formula presented] belongs to [Formula presented]
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