Boundary blow-up solutions and their applications in quasilinear elliptic equations

Based on a comparison principle, we discuss the existence, uniqueness and asymptotic behaviour of various boundary blow-up solutions, for a class of quasilinear elliptic equations, which are then used to obtain a rather complete understanding of some quasilinear elliptic problems on a bounded domain or over the entireR ^N .     

Elliptic Equations with Multi-Singular Inverse-Square Potentials and Critical Nonlinearity

ABSTRACT

This article deals with a class of nonlinear elliptic equations involving a critical power-nonlinearity as well as a potential featuring multiple inverse square singularities. We show that existence of solutions heavily depends on the strength and the location of the singularities. We associate to the problem the corresponding Rayleigh quotient and give both sufficient and necessary conditions on masses and location of singularities for the minimum to be achieved. Both the cases of whole ? ^N and bounded domains are taken into account.     

Existence of multiple positive solutions for −Δu−μ(u/∣x∣2)=u2*−1+σf(x)

In this paper, we consider the semilinear elliptic problem where Ω??^N (N?3) is a bounded smooth domain such that 0∈Ω, σ>0 is a real parameter, and f(x) is some given function in L^∞(Ω) such that f(x)?0, f(x)?0 in Ω. Some existence results of multiple solutions have been obtained by implicit function theorem, monotone iteration method and Mountain Pass Lemma. Copyright © 2002 John Wiley & Sons, Ltd.     

Boundary blow-up solutions for a class of quasilinear elliptic equations

For a given bounded domain Ω in R ^N with smooth boundary ?Ω, we give sufficient conditions on f so that the m-Laplacian equation △ _m u = f(x, u, ?u) admits a boundary blow-up solution u ∈ W ^1,p (Ω). Our main results are new and extend the results in J.V. Concalves and Angelo Roncalli [Boundary blow-up solutions for a class of elliptic equations on a bounded domain, Appl. Math. Comput. 182 (2006), pp. 13–23]. Our approach employs the method of lower–upper solution theorem, fixed point theory and weak comparison principle.     

Pseudo-differential operators with involutive double characteristics

Suppose G is a bounded C²domain in IR ⁿ, n ? 2 . We exam¬ine the regularity at the boundary of solutions to a class of quasi-linear elliptic equations having continuous boundary values ? . If ? has a modulus of continuity β , we give a modulus of continunity for the solution which depends on β and the generalized mean curvature of ?G . When the order of non-uniformity of the equation is between 0 and 1 , no curvature condition on ?G is needed.     

Existence and regularity results for relaxed Dirichlet problems with measure data

We study the following relaxed Dirichlet problem $$\left\{ \begin{gathered} Lu + \mu u = vin\Omega , \hfill \\ u = 0on\partial \Omega , \hfill \\ \end{gathered} \right.$$ where Ω is a bounded open subset ofR ^N,Lu=?div(A?u) is an elliptic operator, μ is a positive Borel measure on Ω not charging polar sets, and v is a measure with bounded variation on Ω. We give a definition of solution for such a problem, and then prove existence and regularity results. As a consequence, the Green function for relaxed Dirichlet problems can be defined, and some of its properties are proved, including the standard representation formula for solutions.     

Self-Similar Solutions of a Convection Diffusion Equation And Related Semilinear Elliptic Problems

We show that for each M>o, and locally Lipschitz function the elliptic equation: in R^N has a positive and exponentially decaying solution with If Ψ is the solution is unique and strictly positive, and if Ψ is the solution is also . Because of the nonvariational nature of the elliptic problem, we use a topological degree argument. The existence of a family of positive self-similar solutions of the parabolic equation in x R^N with follows. They are “source-type” solutions of the convection-diffusion equation above.     

Uniqueness for the two-dimensional Navier–Stokes equation with a measure as initial vorticity

We show that any solution of the two-dimensional Navier-Stokes equation whose vorticity distribution is uniformly bounded in L¹(R²) for positive times is entirely determined by the trace of the vorticity at t=0, which is a finite measure. When combined with previous existence results by Cottet, by Giga, Miyakawa & Osada, and by Kato, this uniqueness property implies that the Cauchy problem for the vorticity equation in R² is globally well-posed in the space of finite measures. In particular, this provides an example of a situation where the Navier-Stokes equation is well-posed for arbitrary data in a function space that is large enough to contain the initial data of some self-similar solutions.     

Large solutions of coupled sublinear/superlinear elliptic equations

We show that large positive solutions exist for the semilinear elliptic equation Δu = p(x)u ^α + q(x)v ^β on bounded domains in R ⁿ , n ≥ 3, for the superlinear case 0 < α ≤ β, β > 1, but not the sublinear case 0 < α ≤ β ≤ 1. We also show that entire large positive solutions exist for both the superlinear and sublinear cases provided the nonnegative continuous functions p and q satisfy certain decay conditions at infinity. Existence and nonexistence of entire bounded solutions are established as well.     

Infinitely many solutions for elliptic problems with variable exponent and nonlinear boundary conditions

We analyze a class of quasilinear elliptic problems involving a p(·)-Laplace-type operator on a bounded domain W ì \mathbb R^N{\Omega\subset{\mathbb R}^N}, N ≥ 2, and we deal with nonlinear conditions on the boundary. Working on the variable exponent Lebesgue–Sobolev spaces, we follow the steps described by the “fountain theorem” and we establish the existence of a sequence of weak solutions.     

Existence of three nontrivial solutions for semilinear elliptic equations on ℝ<Superscript><Emphasis Type="Italic">N</Emphasis></Superscript>

We establish the existence theorem of three nontrivial solutions for a class of semilinear elliptic equation on ? ^N by using variational theorems of mixed type due to Marino and Saccon and linking theorem.     

Multiple positive radial solutions on annuli for nonlinear Neumann problems with large growth

We consider a classical semilinear elliptic equation with Neumann boundary conditions on an annulus in R ^N . The nonlinear term is the product of a radially symmetric coefficient with a pure power. We prove that if the power is sufficiently large, the problem admits at least three distinct positive and radial solutions. In case the coefficient is constant, we show that none of the three solutions is constant. The methods are variational and are based on the study of a suitable limit problem.     

Remark on asymptotic behaviors of solutions to a quasi-linear elliptic Dirichlet problem with large diffusion

We study asymptotic behaviors of nontrivial solutions to the Dirichlet problem of a quasi-linear elliptic equation and obtain a lower bound for growth of L^∞-norm of the solutions, which implies the L^∞-norm of the solutions goes to infinity as the diffusion coefficient goes to infinity.     

Existence and asymptotic behavior of radially symmetric ground states of quasi-linear singular elliptic equations

Existence and asymptotic behavior of entire positive solutions of a class of quasi-linear elliptic equation is obtained. Under several hypotheses on the ρ(x) and f(r), we obtain the existence of positive entire solution. Asymptotic behavior is discussed by constructing an upper solution. The results of this paper is new and extend previously known results.     

Symmetry of solutions for quasimonotone second-order elliptic systems in ordered Banach spaces

We consider symmetry properties of solutions to nonlinear elliptic boundary value problems defined on bounded symmetric domains of \mathbb Rⁿ{\mathbb R^n} . The solutions take values in ordered Banach spaces E, e.g. E=\mathbb R^N{E=\mathbb R^N} ordered by a suitable cone. The nonlinearity is supposed to be quasimonotone increasing. By considering cones that are different from the standard cone of componentwise nonnegative elements we can prove symmetry of solutions to nonlinear elliptic systems which are not covered by previous results. We use the method of moving planes suitably adapted to cover the case of solutions of nonlinear elliptic problems with values in ordered Banach spaces.     

A new perturbation to a critical elliptic problem with a variable exponent

In this paper, we study the following critical elliptic problem with a variable exponent:■,where ■, and ? is a smooth bounded domain in R^N(N≥4). We show that for small enough, there exists a family of bubble solutions concentrating at the negative stable critical point of the function a(x). This is a new perturbation to the critical elliptic equation in contrast to the usual subcritical or supercritical perturbation, and gives the first existence result for the critical elliptic probl...     

Weighted Hardy's inequality and the Kolmogorov equation perturbed by an inverse-square potential

In this article, we give necessary and sufficient conditions for the existence of a weak solution of a Kolmogorov equation perturbed by an inverse-square potential. More precisely, using a weighted Hardy's inequality with respect to an invariant measure μ, we show the existence of the semigroup solution of the parabolic problem corresponding to a generalized Ornstein–Uhlenbeck operator perturbed by an inverse-square potential in L ²(? ^N ,?μ). In the case of the classical Ornstein–Uhlenbeck operator we obtain nonexistence of positive exponentially bounded solutions of the parabolic problem if the coefficient of the inverse-square function is too large.     

Local solvability of a nonstationary leakage problem for an ideal incompressible fluid, 3

In this paper we prove the existence and uniqueness of solutions of the leakage problem for the Euler equations in bounded domain Ω C R³ with corners π/n, n = 2, 3… We consider the case where the tangent components of the vorticity vector are given on the part S₁ of the boundary where the fluid enters the domain. We prove the existence of an unique solution in the Sobolev space W_p^l(Ω), for arbitrary natural l and p > 1. The proof is divided on three parts: (1) the existence of solutions of the elliptic problem in the domain with corners where v – velocity vector, ω – vorticity vector and n is an unit outward vector normal to the boundary, (2) the existence of solutions of the following evolution problem for given velocity vector (3) the method of successive approximations, using solvability of problems (1) and (2).     

Equivalence between entropy and renormalized solutions for parabolic equations with smooth measure data

We consider the nonlinear heat equation (with Leray-Lions operators) on an open bounded subset of R^N with Dirichlet homogeneous boundary conditions. The initial condition is in L¹ and the right hand side is a smooth measure. We extend a previous notion of entropy solutions and prove that they coincide with the renormalized solutions.     

On Blow Up Solutions of a Quasilinear Elliptic Equation

Given a bounded regular domain Ω in ℝ^N, we study existence and asymptotic behaviour of the solutions of the equation Δu + |Du|^q = f(u) in Ω, which diverge on ∂Ω. We extend and complete some results contained in [4].