Existence and uniqueness for a class of quasilinear elliptic boundary value problems

We prove existence and uniqueness of positive solutions for the boundary value problem

(rN−1φ(u′))′=−λrN−1f(u),u′(0)=u(1)=0,     

Existence and uniqueness theorems for singular anisotropic quasilinear elliptic boundary value problems

On bounded domains we consider the anisotropic problems in with 1$"> and on and in with and on . Moreover, we generalize these boundary value problems to space-dimensions 2$">. Under geometric conditions on and monotonicity assumption on we prove existence and uniqueness of positive solutions.     

Sign-changing solutions of a class of nonlocal quasilinear elliptic boundary value problems

This paper is concerned with nonlocal Kirchhoff?s equation

     

Free boundary and mixed boundary value problems for quasilinear elliptic equations: tangential oblique derivative and degenerate Dirichlet boundary problems

We present Hölder estimates and Hölder gradient estimates for a class of free boundary problems with tangential oblique derivative boundary conditions provided the oblique vector β does not vanish at any point on the boundary. We also establish the existence result for a general class of quasilinear degenerate problems of this type including nonlinear wave systems and the unsteady transonic small disturbance equation.     

Applications of differential calculus to quasilinear elliptic boundary value problems with non-smooth data

This paper concerns boundary value problems for quasilinear second order elliptic systems which are, for example, of the type

Transition-layer solutions of quasilinear elliptic boundary blow-up problems and dirichlet problems

We study profiles of positive solutions for quasilinear elliptic boundary blow-up problems and Dirichlet problems with the same equation:

Superlinear elliptic boundary value problems

Summary We use a new form of the mountain pass lemma to extend results on superlinear boundary value problems. Research supported in part by an NSF grant This article was processed by the author using the LATEX style filecljour1m from Springer-Verlag.     

Degenerate elliptic boundary value problems

In this paper we find conditions that guarantee that regular boundary value problems for elliptic differential-operator equations of the second order in an interval are coercive and Fredholm, and we prove the compactness of a resolvent. We apply this result to find some algebraic conditions that guarantee that regular boundary value problems for degenerate elliptic differential equations of the second order in cylindrical domains have the same properties. Note that considered boundary value conditions are nonlocal and are differential only in their principal part, and a domain is nonsmooth.     

Constant boundary value problem for a quasilinear elliptic system

We study the constant boundary value problem for a class of quasilinear elliptic systems. The existence of a nonconstant solution is proved by discussing the minimization problem for the relaxed energy. As a by-product, we show that generally the Pohozaev identity does not hold for H¹-weak solutions.     

Singularly perturbed elliptic boundary value problems

Summary The paper treats elliptic operators of the form L(ɛ∂₁, ..., ɛ∂_n), where L is a polynomial in a variables of order 2m₁, and ɛ is a small parameter. Solutionsu _ɛ of Lu=0 in a half space satisfyng conditions B_j(ɛ∂₁, ɛ∂₂, ..., ɛ∂_n)u=ɛγ_jϕ_j(x)(j=1, ..., m₁) on the boundary are constructed and estimated using H?lder norms, Poisson kernels, and an elaborate potential theory. Properties of the interior limit u₀=u _ɛ(κ) are studied. The paper is preparatory to a detailed investigation of Schauder estimates for such problems with variables coefficients. Supported in part by N. S. F. Grant GP-11660. Entrata in Redazione il 9 gennaio 1971.     

Partially overdetermined elliptic boundary value problems

We consider semilinear elliptic Dirichlet problems in bounded domains, overdetermined with a Neumann condition on a proper part of the boundary. Under different kinds of assumptions, we show that these problems admit a solution only if the domain is a ball. When these assumptions are not fulfilled, we discuss possible counterexamples to symmetry. We also consider Neumann problems overdetermined with a Dirichlet condition on a proper part of the boundary, and the case of partially overdetermined problems on exterior domains.     

Asymptotically linear elliptic boundary value problems

We study semilinear problems in which the nonlinear term has different asymptotic behavior at ± with the limits (1.2) spanning a finite number of eigenvalues of the linear operator.Research supported in part by an NSF grant.     

On fourth-order elliptic boundary value problems

This paper is concerned with the existence and uniqueness of a solution for a class of fourth-order elliptic boundary value problems. The existence of a solution is proven by the method of upper and lower solutions without any monotone nondecreasing or nonincreasing property of the nonlinear function. Sufficient conditions for the uniqueness of a solution and some techniques for the construction of upper and lower solutions are given. All the existence and uniqueness results are directly applicable to fourth-order two-point boundary value problems.

     

Nonselfadjoint semilinear elliptic boundary value problems

Summary In this paper we study the existence of solutions of nonselfadjoint semilinear elliptic boundary value problems with a bounded nonlinear term. We emphasize that this nonlinear term may depend on the derivatives of the function in a nontrivial way. In the proof of our main result we use the Leray-Schauder degree theory.Supported in part by the C.A.I.C.Y.T., Ministry of Education (Spain), under Grant no. 3258/83.     

First boundary value problem for systems of quasilinear elliptic equations

One proves the solvability of the Dirichlet problem for a certain classes of degenerate, quasilinear, second-order elliptic system. This is done with the aid of operator multiplicative functionals of Markov process.