Large solution to nonlinear elliptic equation with nonlinear gradient terms

The aim of this paper is to study the qualitative behavior of large solutions to the following problem

     

On uniqueness for nonlinear elliptic equation involving the Pucci's extremal operator

In this article we study uniqueness of positive solutions for the nonlinear uniformly elliptic equation in RN, limr→∞u(r)=0, where denotes the Pucci's extremal operator with parameters 0<λ?Λ and p>1. It is known that all positive solutions of this equation are radially symmetric with respect to a point in RN, so the problem reduces to the study of a radial version of this equation. However, this is still a nontrivial question even in the case of the Laplacian (λ=Λ). The Pucci's operator is a prototype of a nonlinear operator in no-divergence form. This feature makes the uniqueness question specially challenging, since two standard tools like Pohozaev identity and global integration by parts are no longer available. The corresponding equation involving is also considered.     

A boundary blow-up elliptic problem with an inhomogeneous term

By a perturbation method and constructing comparison functions, we reveal how the inhomogeneous term h$h$ affects the exact asymptotic behaviour of solutions near the boundary to the problem △u=b(x)g(u)+λh(x)$△u=b\left(x\right)g\left(u\right)+\lambda h\left(x\right)$, u>0$u>0$ in Ω$\Omega$, u_|∂Ω=∞$u{|}_{\partial \Omega }=\infty$, where Ω$\Omega$ is a bounded domain with smooth boundary in R^N${\mathbb{R}}^N$, λ>0$\lambda >0$, g∈C¹[0,∞)$g\in C^1[0,\infty )$ is increasing on [0,∞)$[0,\infty )$, g(0)=0$g\left(0\right)=0$, g^′$g^{\prime }$ is regularly varying at infinity with positive index ρ$\rho$, the weight b$b$, which is non-trivial and non-negative in Ω$\Omega$, may be vanishing on the boundary, and the inhomogeneous term h$h$ is non-negative in Ω$\Omega$ and may be singular on the boundary.     

Positive radial solutions to a ‘semilinear’ equation involving the Pucci's operator

In this article we prove existence of positive radially symmetric solutions for the nonlinear elliptic equation

     

Monotonicity of solutions of Fully nonlinear uniformly elliptic equations in the half-plane

In this paper we study the monotonicity of positive (or non-negative) viscosity solutions to uniformly elliptic equations F(∇u,D²u)=f(u) in the half plane, where f is locally Lipschitz continuous (with f(0)?0) and zero Dirichlet boundary conditions are imposed. The result is obtained without assuming the u or |∇u| are bounded.     

Multiple positive solutions for a quasilinear elliptic equation with critical exponential nonlinearity

Let Ω be a bounded domain in R^N,N≥2, with C² boundary. In this work, we study the existence of multiple positive solutions of the following problem:

     

An irregular complex valued solution to a scalar uniformly elliptic equation

We present an irregular weak solution of a uniformly elliptic scalar equation in divergence form with measurable coefficients. The solution has a square integrable gradient. Such examples have been known for dimension n ≥ 5 only. The author was partially supported by SFB 611.     

Existence and nonexistence of ground state solutions for elliptic equations with a convection term

We deal with the existence of positive solutions u decaying to zero at infinity, for a class of equations of Lane-Emden-Fowler type involving a gradient term. One of the main points is that the differential equation contains a semilinear term σ(u) where σ:(0,∞)→(0,∞) is a smooth function which can be both unbounded at infinity and singular at zero. Our technique explores symmetry arguments as well as lower and upper solutions.     

On a semilinear elliptic equation with inverse-square potential

We study the existence and nonexistence of solutions to a semilinear elliptic equation with inverse-square potential. The dividing line with respect to existence or nonexistence is given by a critical exponent, which depends on the strength of the potential.     

Quasilinear elliptic system with coupling on nonhomogeneous critical term

We study the existence of multiple solutions for a quasilinear elliptic system of gradient type with the possibility of coupling on the critical and subcritical terms which are not necessarily homogeneous. The solutions are obtained from a version of the Symmetric Mountain Pass Theorem. A version of the Concentration-Compactness Principle for this class of systems allows us to verify that the Palais–Smale condition is satisfied below a certain level.     

Multiplicity of solutions to uniformly elliptic fully nonlinear equations with concave-convex right-hand side

We deal with existence, non-existence and multiplicity of solutions to the model problem

(P)     

Infinitely many solutions to perturbed elliptic equations

A new version of perturbation theory is developed which produces infinitely many sign-changing critical points for uneven functionals. The abstract result is applied to the following elliptic equations with a Hardy potential and a perturbation from symmetry:

     

Degree and global bifurcation for elliptic equations with multivalued unilateral conditions

A bifurcation problem for an elliptic multivalued boundary value problem with a real parameter is considered. The existence of global bifurcation between two eigenvalues of a certain type of the Laplacian is proved. For a class of abstract inclusions with compact multivalued mappings in a Hilbert space, it is shown how the degree can be determined near the eigenvalues of a particular type of an associated linear single-valued problem, and the jump of the degree is proved. As a consequence, global bifurcation for such abstract inclusions is obtained. The weak formulation of the boundary value problem mentioned is a particular case.     

Multiple solutions for elliptic systems with nonlinearities of arbitrary growth

We prove the existence of infinitely many solutions for symmetric elliptic systems with nonlinearities of arbitrary growth. Moreover, if the symmetry of the problem is broken by a small enough perturbation term, we find at least three solutions. The proofs utilise a variational setting given by de Figueiredo and Ruf in order to prove an existence's result and the “algebraic” approach based on the Pohozaev's fibering method.     

Compactness of sequences of two-dimensional energies with a zero-order term. Application to three-dimensional nonlocal effects

In this paper we study the limit, in the sense of the Γ-convergence, of sequences of two-dimensional energies of the type , where A _n is a symmetric positive definite matrix-valued function and μ _n is a nonnegative Borel measure (which can take infinite values on compact sets). Under the sole equicoerciveness of A _n we prove that the limit energy belongs to the same class, i.e. its reads as , where is a diffusion independent of μ _n and μ is a nonnegative Borel measure which does depend on . This compactness result extends in dimension two the ones of [11,23] in which A _n is assumed to be uniformly bounded. It is also based on the compactness result of [7] obtained for sequences of two-dimensional diffusions (without zero-order term). Our result does not hold in dimension three or greater, since nonlocal effects may appear. However, restricting ourselves to three-dimensional diffusions with matrix-valued functions only depending on two coordinates, the previous two-dimensional result provides a new approach of the nonlocal effects. So, in the periodic case we obtain an explicit formula for the limit energy specifying the kernel of the nonlocal term.     

A remark on entire solutions of quasilinear elliptic equations

By applying a main comparison theorem of Pucci and Serrin (2007) [2] we cover, for general equations of p-Laplace type, the open cases of Theorems B, D, E of Farina and Serrin (submitted for publication) [1] as described in Problems 2 and 3 of Section 12 of Farina and Serrin (submitted for publication) [1]. Moreover, we provide significant improvements of Theorem C and Theorem 5 of Farina and Serrin (submitted for publication) [1], the latter in the context of mean curvature type operators, see Theorem 1.3 and Theorems 5.2-5.4 below.Finally, Theorem 1.1 provides a new Liouville theorem outside the context of work in Farina and Serrin (submitted for publication) [1].     

Boundary behaviour for solutions of elliptic singular equations with a gradient term

We investigate the homogeneous Dirichlet boundary value problem for a class of second-order nonlinear elliptic partial differential equations with a gradient term and singular data. Under general conditions on the data, we study the behaviour of the solution near the boundary of the domain. Under suitable additional conditions we also investigate the second-order term in the asymptotic expansion of the solution in terms of the distance from the boundary.