Existence of solutions to nonlinear, subcritical higher order elliptic Dirichlet problems

We consider the 2m-th order elliptic boundary value problem Lu=f(x,u) on a bounded smooth domain Ω⊂RN with Dirichlet boundary conditions on ∂Ω. The operator L is a uniformly elliptic linear operator of order 2m whose principle part is of the form . We assume that f is superlinear at the origin and satisfies , , where are positive functions and q>1 is subcritical. By combining degree theory with new and recently established a priori estimates, we prove the existence of a nontrivial solution.     

Uniqueness and exact multiplicity of solutions for a class of Dirichlet problems

We apply the “monotone separation of graphs” technique of L.A. Peletier and J. Serrin [L.A. Peletier, J. Serrin, Uniqueness of positive solutions of semilinear equations in Rn, Arch. Ration. Mech. Anal. 81 (2) (1983) 181-197; L.A. Peletier, J. Serrin, Uniqueness of nonnegative solutions of semilinear equations in Rn, J. Differential Equations 61 (3) (1986) 380-397], as developed further by L. Erbe and M. Tang [L. Erbe, M. Tang, Structure of positive radial solutions of semilinear elliptic equations, J. Differential Equations 133 (2) (1997) 179-202], to the question of exact multiplicity of positive solutions for a class of semilinear equations on a unit ball in Rn. We also observe that using P. Pucci and J. Serrin [P. Pucci, J. Serrin, Uniqueness of ground states for quasilinear elliptic operators, Indiana Univ. Math. J. 47 (2) (1998) 501-528] improvement of a certain identity of L. Erbe and M. Tang [L. Erbe, M. Tang, Structure of positive radial solutions of semilinear elliptic equations, J. Differential Equations 133 (2) (1997) 179-202] produces a short proof of L. Erbe and M. Tang [L. Erbe, M. Tang, Structure of positive radial solutions of semilinear elliptic equations, J. Differential Equations 133 (2) (1997) 179-202] result on the uniqueness of positive solution of (1<p, )

     

Bubble towers for supercritical semilinear elliptic equations

We construct positive solutions of the semilinear elliptic problem with Dirichet boundary conditions, in a bounded smooth domain Ω⊂R^N(N?4), when the exponent p is supercritical and close enough to and the parameter λ∈R is small enough. As , the solutions have multiple blow up at finitely many points which are the critical points of a function whose definition involves Green's function. Our result extends the result of Del Pino et al. (J. Differential Equations 193(2) (2003) 280) when Ω is a ball and the solutions are radially symmetric.     

Ground state solutions for a semilinear elliptic problem with a convection term

By a sub-supersolution method and a perturbed argument, we improve the earlier results concerning the existence of ground state solutions to a semilinear elliptic problem −Δu+p(x)q|∇u|=f(x,u), u>0, x∈RN, , where q∈(1,2], for some α∈(0,1), p(x)?0, ∀x∈RN, and f:RN×(0,∞)→[0,∞) is a locally Hölder continuous function which may be singular at zero.     

On upper bounds for positive solutions of semilinear equations

Suppose that E is a bounded domain of class C^2,λ in and L is a uniformly elliptic operator in E. The set of all positive solutions of the equation Lu=ψ(u) in E was investigated by a number of authors for various classes of functions ψ. In Dynkin and Kuznetsov (Comm. Pure Appl. Math. 51 (1998) 897) we defined, for every Borel subset Γ of ∂E, two such solutions u_Γ?w_Γ. We also introduced a class of solutions u_ν in 1-1 correspondence with a certain class of σ-finite measures ν on ∂E. With every we associated a pair (Γ,ν) where Γ is a Borel subset of ∂E and . We called this pair the fine boundary trace of u and we denoted in tr(u).Let u⊕v stand for the maximal solution dominated by u+v. We say that u belongs to the class if the condition tr(u)=(Γ,ν) implies that u?w_Γ⊕u_ν and we say that u belongs to if the condition tr(u)=(Γ,ν) implies that u?u_Γ⊕u_ν.It was proved in Dynkin and Kuznetsov (1998) that, under minimal assumptions on L and ψ, the class contains all bounded domains. It follows from results of Mselati (Thése de Doctorat de l'Université Paris 6, 2002; C.R. Acad. Sci. Paris Sér. I 332 (2002); Mem. Amer. Math. Soc. (2003), to appear), that all E of the class C⁴ belong to where Δ is the Laplacian and ψ(u)=u². [Mselati proved that, in his case, u_Γ=w_Γ and therefore the condition tr(u)=(Γ,ν) implies u=u_Γ⊕u_ν=w_Γ⊕u_ν.]By modifying Mselati's arguments, we extend his result to ψ(u)=u^α with 1<α?2 and all bounded domains of class C^2,λ.We start from proving a general localization theorem: under broad assumptions on L, ψ if, for every y∈∂E there exists a domain such that E′⊂E and ∂E∩∂E′ contains a neighborhood of y in ∂E.     

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.     

Sublinear singular elliptic problems with two parameters

We establish several existence and nonexistence results for the boundary value problem −Δu+K(x)g(u)=λf(x,u)+μh(x) in Ω, u=0 on ∂Ω, where Ω is a smooth bounded domain in , λ and μ are positive parameters, h is a positive function, while f has a sublinear growth. The main feature of this paper is that the nonlinearity g is assumed to be unbounded around the origin. Our analysis shows the importance of the role played by the decay rate of g combined with the signs of the extremal values of the potential K(x) on . The proofs are based on various techniques related to the maximum principle for elliptic equations.     

Relaxation and regularity in the calculus of variations

In this work we prove that, if L(t,u,ξ) is a continuous function in t and u, Borel measurable in ξ, with bounded non-convex pieces in ξ, then any absolutely continuous solution to the variational problem

     

Uniqueness of positive radial solutions for Δu−u+u=0 on an annulus

We prove uniqueness of positive radial solutions to the semilinear elliptic equation , subject to the Dirichlet boundary condition on an annulus in . As a by-product, our argument also provides a much simpler, if not the simplest, new proof for the uniqueness of positive solutions to the same problem in a finite ball or in the whole space .     

Nonlinear elliptic systems with general growth

We prove local Lipschitz-continuity and, as a consequence, C^kandC^∞regularity of weak solutions u for a class of nonlinear elliptic differential systems of the form . The growth conditions on the dependence of functions on the gradient Du are so mild to allow us to embrace growths between the linear and the exponential cases, and they are more general than those known in the literature.     

The Liouville equation in a half-plane

We classify the solutions of the equation Δu+aeu=0 in the half-plane that satisfy the Neumann boundary condition ∂u/∂t=ceu/2 on . An analogous problem in the once punctured disc D^∗⊂R² is also solved.     

Up-to boundary regularity for a singular perturbation problem of p-Laplacian type

In this paper we are interested in establishing up-to boundary uniform estimates for the one phase singular perturbation problem involving a nonlinear singular/degenerate elliptic operator. Our main result states: if Ω⊂Rn is a C1,α domain, for some 0<α<1 and uε verifies

     

Multiplicity for strongly indefinite semilinear elliptic system

In this paper we study a multiplicity result for a strongly indefinite semilinear elliptic system

     

A minimum problem with free boundary in Orlicz spaces

We consider the optimization problem of minimizing in the class of functions W1,G(Ω) with , for a given φ₀?0 and bounded. W1,G(Ω) is the class of weakly differentiable functions with . The conditions on the function G allow for a different behavior at 0 and at ∞. We prove that every solution u is locally Lipschitz continuous, that it is a solution to a free boundary problem and that the free boundary, Ω∩∂{u>0}, is a regular surface. Also, we introduce the notion of weak solution to the free boundary problem solved by the minimizers and prove the Lipschitz regularity of the weak solutions and the C1,α regularity of their free boundaries near “flat” free boundary points.