Positive solutions for singular elliptic equations with mixed Dirichlet‐Neumann boundary conditions

In this paper, we investigate the existence of positive solutions for singular elliptic equations with mixed Dirichlet‐Neumann boundary conditions involving Sobolev‐Hardy critical exponents and Hardy terms by using the concentration compactness principle, the strong maximum principle and the Mountain Pass lemma. We also prove, under complementary conditions, that there is no nontrivial solution if the domain is star‐shaped with respect to the origin.     

Quasilinear elliptic equations with boundary blow-up

Assume that Ω is a bounded domain in ℝ _N withN ≥2, which has aC ²-boundary. We show that forp ∃ (1, ∞) there exists a weak solutionu of the problem δ_p u(x) = f(u(x)), x ∃ Ω with boundary blow-up, wheref is a positive, increasing function which meets some natural conditions. The boundary blow-up ofu(x) is characterized in terms of the distance ofx from ∂Ω. For the Laplace operator, our results coincide with those of Bandle and Essén [1]. Finally, for a rather wide subclass of the class of the admissible functionsf, the solution is unique whenp ∃ (1, 2].     

Positive solutions for critical quasilinear elliptic equations with mixed dirichlet-neumann boundary conditions

The existence and multiplicity of positive solutions are studied for a class of quasilinear elliptic equations involving Sobolev critical exponents with mixed Dirichlet-Neumann boundary conditions by the variational methods and some analytical techniques.     

Quasilinear elliptic equations with subquadratic growth

In this paper we consider nonlinear boundary value problems whose simplest model is the following:

(0.1)     

Quasilinear elliptic equations with small boundary data

It is shown that if the order of non-uniformity of a quasilinear elliptic equation is h, 1<h2, then the critical norm separating existence and non-existence of a solution to the Dirichlet problem with small boundary data is the norm. For 0h1, existence of a solution is guaranteed without any smallness assumption on the given boundary data, provided that the usual a priori interior gradient bound for solution is available.     

Critical Exponents of Quasilinear Parabolic Equations

In this paper we study the critical exponents of the Cauchy problem in Rⁿ of the quasilinear singular parabolic equations: u_t = div(|∇u|^m − 1∇u) + t^s|x|^σu^p, with non-negative initial data. Here s ≥ 0, (n − 1)/(n + 1) < m < 1, p > 1 and σ > n(1 − m) − (1 + m + 2s). We prove that p_c ≡ m + (1 + m + 2s + σ)/n > 1 is the critical exponent. That is, if 1 < p ≤ p_c then every non-trivial solution blows up in finite time, but for p > p_c, a small positive global solution exists.     

Quasilinear elliptic inequalities on complete Riemannian manifolds

We prove maximum and comparison principles for weak distributional solutions of quasilinear, possibly singular or degenerate, elliptic differential inequalities in divergence form on complete Riemannian manifolds. A new definition of ellipticity for nonlinear operators on Riemannian manifolds is introduced, covering the standard important examples. As an application, uniqueness results for some related boundary value problems are presented.     

Competition reaction-absorption in some elliptic and parabolic problems related to the Caffarelli-Kohn-Nirenberg inequalities

We consider the following elliptic problem:

(1)     

Quasilinear parabolic and elliptic equations with nonlinear boundary conditions

This paper is concerned with a class of quasilinear parabolic and elliptic equations in a bounded domain with both Dirichlet and nonlinear Neumann boundary conditions. The equation under consideration may be degenerate or singular depending on the property of the diffusion coefficient. The consideration of the class of equations is motivated by some heat-transfer problems where the heat capacity and thermal conductivity are both temperature dependent. The aim of the paper is to show the existence and uniqueness of a global time-dependent solution of the parabolic problem, existence of maximal and minimal steady-state solutions of the elliptic problem, including conditions for the uniqueness of a solution, and the asymptotic behavior of the time-dependent solution in relation to the steady-state solutions. Applications are given to some heat-transfer problems and an extended logistic reaction–diffusion equation.     

Multiple solutions for semilinear elliptic equations with Neumann boundary condition and jumping nonlinearities

We obtain nonconstant solutions of semilinear elliptic Neumann boundary value problems with jumping nonlinearities when the asymptotic limits of the nonlinearity fall in the type (Il), l>2 and (IIl), l?1 regions formed by the curves of the Fucik spectrum. Furthermore, we have at least two nonconstant solutions in every order interval under resonance case. In this paper, we apply the sub-sup solution method, Fucik spectrum, mountain pass theorem in order intervals, degree theory and Morse theory to get the conclusions.     

Quasilinear second order elliptic equations with Venttsel boundary conditions

We prove the existence of solutions of quasilinear second order elliptic equations with quasilinear Venttsel boundary conditions in both classical and weak sense, under natural structure conditions.     

Positive solutions of linear elliptic equations with critical growth in the Neumann boundary condition

We study the existence of positive solutions of a linear elliptic equation with critical Sobolev exponent in a nonlinear Neumann boundary condition. We prove a result which is similar to a classical result of Brezis and Nirenberg who considered a corresponding problem with nonlinearity in the equation. Our proof of the fact that the dimension three is critical uses a new Pohoaev-type identity.AMS Subject Classification: Primary: 35J65; Secondary: 35B33.     

Hardy-Sobolev critical singular elliptic equations with mixed Dirichlet-Neumann boundary conditions

Three important inequalities (the Poincaré, Hardy and generalized Poincaré inequalities) on the mixed boundary conditions are firstly established by some analytical techniques. Then the existence and multiplicity of positive solutions are studied for a class of semilinear elliptic equations with mixed Dirichlet-Neumann boundary conditions involving Hardy terms and Hardy-Sobolev critical exponents by using the variational methods.     

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.     

Existence result for a class of singular quasilinear elliptic equations with critical growth

In this paper, we obtain the existence of a nontrivial solution for a class of singular quasilinear elliptic equations with critical exponents. The proofs rely on a non-smooth critical point theory, and some techniques used by Brezis and Nirenberg.     

The Neumann problem for some degenerate elliptic equations

In the paper we study the equation L _u = f, where L is a degenerate elliptic operator, with Neumann boundary condition in a bounded open set μ. We prove existence and uniqueness of solutions in the space H(μ) for the Neumann problem.     

On bi-nonlocal problem for elliptic equations with Neumann boundary conditions

We prove the existence of positive solutions for a nonlocal problem (1.2) with Neumann boundary conditions. We distinguish two cases: 2 < p < 2* (subcritical) and p = 2* (critical). The existence of solutions is established by variational methods.     

Classification and non-existence results for weak solutions to quasilinear elliptic equations with Neumann or Robin boundary conditions

We classify positive solutions to a class of quasilinear equations with Neumann or Robin boundary conditions in convex domains. Our main tool is an integral formula involving the trace of some relevant quantities for the problem.Under a suitable condition on the nonlinearity, a relevant consequence of our results is that we can extend to weak solutions a celebrated result obtained for stable solutions by Casten and Holland and by Matano.