Positive weak solutions of semilinear second order elliptic inequalities via variational inequalities

Existence, uniqueness and convergence of approximants of positive weak solutions for semilinear second order elliptic inequalities are obtained. The nonlinearities involved in these inequalities satisfy suitable upper or lower bound conditions or monotonicity conditions. The lower bound conditions are allowed to contain the critical Sobolev exponents. The methodology is to establish variational inequality principles for demicontinuous pseudo-contractive maps in Hilbert spaces by considering convergence of approximants and apply them to the corresponding variational inequalities arising from the semilinear second order elliptic inequalities. Examples on the existence, uniqueness and convergence of approximants of positive weak solutions of the semilinear second order elliptic inequalities are given.     

On bifurcation for semilinear elliptic Dirichlet problems on geodesic balls

We study bifurcation from a branch of trivial solutions of semilinear elliptic Dirichlet boundary value problems on a geodesic ball, whose radius is used as the bifurcation parameter. In the proof of our main theorem we obtain in addition a special case of an index theorem due to S. Smale.     

The existence of solutions for a class of semilinear elliptic systems

In this paper, some new existence theorems of weak solutions for a class of semilinear elliptic systems are obtained by means of the local linking theorem and the saddle point theorem.     

Second-order semilinear elliptic inequalities in exterior domains

We study the problem of existence and nonexistence of positive solutions of the semilinear elliptic inequalities in divergence form with measurable coefficients in exterior domains in . For W(x)?|x|^−σ at infinity we compute the critical line on the plane (p,σ), which separates the domains of existence and nonexistence, and reveal the class of potentials V that preserves the critical line. Example are provided showing that the class of potentials is maximal possible, in certain sense. The case of (p,σ) on the critical line has also been studied.     

kth Order convergence for a semilinear elliptic boundary value problem in the divergence form

The convergence problem of approximate solutions for a semilinear elliptic boundary value problem in the divergence form is studied. By employing the method of quasilinearization, a sequence of approximate solutions converging with the kth (k ? 2) order convergence to a weak solution for a semilinear elliptic problem is obtained via the variational approach.     

Some existence results for a class of degenerate semilinear elliptic systems

Using a variational approach, we investigate a class of degenerate semilinear elliptic systems with measurable, unbounded nonnegative weights, where the domain is bounded or unbounded. Some existence results are obtained.     

A class of new iterative methods for general mixed variational inequalities

In this paper, we use the auxiliary principle technique to suggest a class of predictor-corrector methods for solving general mixed variational inequalities. The convergence of the proposed methods only requires the partially relaxed strongly monotonicity of the operator, which is weaker than co-coercivity. As special cases, we obtain a number of known and new results for solving various classes of variational inequalities and related problems.     

Perturbation methods in semilinear elliptic problems involving critical hardy-sobolev exponent

In this article, we deal with a class of semilinear elliptic equations which are perturbations of the problems with the critical Hardy-Sobolev exponent. Some existence results are given via an abstract perturbation method in critical point theory.     

On semilinear elliptic equations with sublinear and superlinear nonlinearities in rN

The existence of infinitely many solutions for the equation     

Multiple solutions for semilinear elliptic boundary value problems with double resonance

In this paper we study the multiplicity of nontrivial solutions of semilinear elliptic boundary value problems which may be double resonance near infinity between two consecutive eigenvalues of −Δ with zero Dirichlet boundary data. The methods we use here are Morse theory, minimax methods and bifurcation theory.     

Multiplicity of solutions for nonlinear second order impulsive differential equations with linear derivative dependence via variational methods

This paper uses critical point theory and variational methods to investigate the multiple solutions of boundary value problems for second order impulsive differential equations. The conditions for the existence of multiple solutions are established. An example is constructed to illustrate the proposed result.     

On well-posedness of semilinear parabolic and elliptic problems in the hyperbolic space

We investigate existence and uniqueness of solutions to semilinear parabolic and elliptic equations in bounded domains of the n-dimensional hyperbolic space (n?3). Lp→Lq estimates for the semigroup generated by the Laplace-Beltrami operator are obtained and then used to prove existence and uniqueness results for parabolic problems. Moreover, under proper assumptions on the nonlinear function, we establish uniqueness of positive classical solutions and nonuniqueness of singular solutions of the elliptic problem; furthermore, for the corresponding semilinear parabolic problem, nonuniqueness of weak solutions is stated.     

A global superconvergent $L^{\infty}$-error estimate of mixed finite element methods for semilinear elliptic optimal control problems

In this paper, we discuss the superconvergence of mixed finite element methods for a semilinear elliptic control problem with an integral constraint. The state and costate are approximated by the order $k=1$ Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. Approximation of the optimal control of the continuous optimal control problem will be constructed by a projection of the discrete adjoint state. It is proved that this approximation has convergence order $h^{2}$ in $L^{\infty}$-norm. Finally, a numerical example is given to demonstrate the theoretical results.     

Asymptotic behavior of solutions of semilinear elliptic equations in unbounded domains: Two approaches

In this paper, we study the asymptotic behavior as x₁→+∞ of solutions of semilinear elliptic equations in quarter- or half-spaces, for which the value at x₁=0 is given. We prove the uniqueness and characterize the one-dimensional or constant profile of the solutions at infinity. To do so, we use two different approaches. The first one is a pure PDE approach and it is based on the maximum principle, the sliding method and some new Liouville type results for elliptic equations in the half-space or in the whole space RN. The second one is based on the theory of dynamical systems.     

Solutions of a class of Hamiltonian elliptic systems in

We study the existence of ground state solutions for the following elliptic systems in R^N

where b=(b₁,…,b_N) is a constant vector and HC¹(R^N×R²,R) is nonperiodic in variables x and super-quadratic as |z|→∞. By a recent critical point theorem for strongly indefinite problem, we obtain the existence of at least one ground state solution.     

A variational treatment for general elliptic equations of the flame propagation type: regularity of the free boundary

We develop a variational theory to study the free boundary regularity problem for elliptic operators: Lu=Dj(a_ij(x)Diu)+biui+c(x)u=0$Lu=D_j(a_{ij}(x)D_{i}u)+b_i{u}_i+c(x)u=0$ in {u>0}$\{u>0\}$, 〈a_ij(x)∇u,∇u〉=2$〈a_{ij}(x)\mathrm{\nabla }u,\mathrm{\nabla }u〉=2$ on ∂{u>0}$\partial \{u>0\}$. We use a singular perturbation framework to approximate this free boundary problem by regularizing ones of the form: Luε=βε(uε)$L{u}_{\epsilon }={\beta }_{\epsilon }(u_{\epsilon })$, where βε${\beta }_{\epsilon }$ is a suitable approximation of Dirac delta function δ₀${\delta }_0$. A useful variational characterization to solutions of the above approximating problem is established and used to obtain important geometric properties that enable regularity of the free boundary. This theory has been developed in connection to a very recent line of research as an effort to study existence and regularity theory for free boundary problems with gradient dependence upon the penalization.     

Minimax methods for singular elliptic equations with an application to a jumping problem

A jumping problem for a class of singular semilinear elliptic equations is considered. Minimax methods in the framework of nonsmooth critical point theory are applied.     

Semilinear problems in unbounded domains

As formulated by Silva [E.A. de B.e. Silva, Linking theorems and applications to semilinear elliptic problems at resonance, Nonlinear Anal. 16 (1991) 455-477] and Schechter [M. Schechter, A generalization of the saddle point method with applications, Ann. Polon. Math. 57 (3) (1992) 269-281; M. Schechter, New saddle point theorems, in: Generalized Functions and Their Applications, Varanasi, 1991, Plenum, New York, 1993, pp. 213-219], the sandwich theorem has become a very useful tool in finding critical points of functionals leading to solutions of partial differential equations. In the present paper, this theorem is strengthened to apply to more general situations. We present some applications.     

Existence of multiple positive solutions for second order nonlinear dynamic BVPs by variational methods

The aim of this paper is to employ variational techniques and critical point theory to prove some sufficient conditions for the existence of multiple positive solutions to a nonlinear second order dynamic equation with homogeneous Dirichlet boundary conditions.