Existence of Solutions to a Semilinear Elliptic System through Orlicz-Sobolev Spaces

Using Orlicz-Sobolev spaces and a variant of the Mountain-Pass Lemma of Ambrosetti-Rabinowitz we obtain existence of a (positive) solution to a semilinear system of elliptic equations. The admissible nonlinearities are such that the system is superlinear and subcritical. The Orlicz setting used here allows us to consider nonlinearities which are not (asymptotically) pure powers. Moreover, by an interpolation theorem of Boyd we find an elliptic regularity result in Orlicz-Sobolev spaces. A bootstrapping argument implies that the above mentioned solutions are classical.     

Existence of three solutions for a non-homogeneous Neumann problem through Orlicz-Sobolev spaces

The aim of this paper is to establish a multiplicity result for an eigenvalue non-homogeneous Neumann problem which involves a nonlinearity fulfilling a nonstandard growth condition. Precisely, a recent critical points result for differentiable functionals is exploited in order to prove the existence of a determined open interval of positive eigenvalues for which the problem admits at least three weak solutions in an appropriate Orlicz-Sobolev space.     

Oblique boundary value problems for equations of Monge-Ampère type

We prove the existence of classical solutions of elliptic equations of Monge-Ampère type subject to a semilinear oblique boundary condition which is a perturbation of the Neumann boundary condition. Our techniques also allow us to treat fully nonlinear strictly oblique boundary conditions satisfying a concavity condition. Examples show that the above restrictions on the boundary condition are generally necessary for the existence of classical solutions. Received May 22, 1996 / Accepted April 10, 1997     

Four positive solutions for the semilinear elliptic equation: -\Delta u+u=a(x)u^p+f(x) in {\mathbb R}^N

We consider the existence of positive solutions of the following semilinear elliptic problem in : where , , , and . Under the conditions: 1° for all , 2° as , 3° there exist and such that 4°, we show that (*) has at least four positive solutions for sufficiently small but . Received December 11, 1998 / Accepted July 16, 1999 / Published online April 6, 2000     

On the existence of positive solutions for semilinear elliptic equations with singular lower order coefficients and Dirichlet boundary conditions

We give sufficient conditions for the existence of positive solutions to some semilinear elliptic equations in bounded domains with Dirichlet boundary conditions. We impose mild conditions on the domains and lower order (nonlinear) coefficients of the equations in that the bounded domains are only required to satisfy an exterior cone condition and we allow the coefficients to have singularities controlled by Kato class functions. Our approach uses an implicit probabilistic representation, Schauder's fixed point theorem, and new a priori estimates for solutions of the corresponding linear elliptic equations. In the course of deriving these a priori estimates we show that the Green functions for operators of the form on D are comparable when one modifies the drift term b on a compact subset of D. This generalizes a previous result of Ancona [2], obtained under an condition on b, to a Kato condition on . Received: 21 April 1998 / in final form 26 March 1999     

Some results on a class of nonliner Schrödinger equations

By using a Liapunov-Schmidt reduction we prove an existence result for the nonlinear Schr?dinger equation in where satisfies suitable assumptions. We also provide a necessary condition for the existence of solutions. Received June 7, 1999 / in final form November 10, 1999 / Published online July 20, 2000     

Stationary layered solutions in {\mathbb{R}}^2 for a class of non autonomous Allen-Cahn equations

We consider a class of non autonomous Allen-Cahn equations where is a multiple-well potential and is a periodic, positive, non-constant function. We look for solutions to (0.1) having uniform limits as corresponding to minima of W. We show, via variational methods, that under a nondegeneracy condition on the set of heteroclinic solutions of the associated ordinary differential equation the equation (0.1) has solutions which depends on both the variables x andy. In contrast, when a is constant such nondegeneracy condition is not satisfied and all such solutions are known to depend only on x. Received April 16, 1999 / Accepted October 1, 1999 / Published online June 28, 2000     

Four versus two solutions of semilinear elliptic boundary value problems

This paper concerns the existence of four (or six) solutions of semilinear elliptic boundary value problems provided that two disorderly solutions are known. The results are obtained under very generic conditions. Received: 26 August 2000 / Accepted: 23 February 2001 / Published online: 23 July 2001     

Convex integration with constraints and applications to phase transitions and partial differential equations

We study solutions of first order partial differential relations Du∈K, where u:Ω⊂ℝ ⁿ →ℝ ^m is a Lipschitz map and K is a bounded set in m×n matrices, and extend Gromov’s theory of convex integration in two ways. First, we allow for additional constraints on the minors of Du and second we replace Gromov’s P-convex hull by the (functional) rank-one convex hull. The latter can be much larger than the former and this has important consequences for the existence of ‘wild’ solutions to elliptic systems. Our work was originally motivated by questions in the analysis of crystal microstructure and we establish the existence of a wide class of solutions to the two-well problem in the theory of martensite. Received April 23, 1999 / final version received September 11, 1999     

Weak solutions for nonlocal boundary value problems with low regularity data

This paper concerns the existence and uniqueness of weak solutions for elliptic and parabolic equations under nonlocal boundary conditions, based on maximal regularity. It also gives the positivity of solutions which can be used in monotone iteration methods. As an application, the results are used to discuss some specific nonlocal problems.     

On the uniqueness and structure of solutions to a coupled elliptic system

In this paper, we consider a nonlinear elliptic system which is an extension of the single equation derived by investigating the stationary states of the nonlinear Schrödinger equation. We establish the existence and uniqueness of solutions to the Dirichlet problem on the ball. In addition, the nonexistence of the ground state solutions under certain conditions on the nonlinearities and the complete structure of different types of solutions to the shooting problem are proved.     

Least energy solutions of a critical Neumann problem with a weight

We investigate the effect of the coefficient of the critical nonlinearity for the Neumann problem on the existence of least energy solutions. As a by-product we establish a Sobolev inequality with interior norm. Received: 26 April 2000 / Accepted: 25 February 2001 / Published online: 5 September 2002     

Some interior regularity results for solutions of Hessian equations

We prove monotonicity formulae related to degenerate k-Hessian equations which yield Morrey type estimates for certain integrands involving the second derivatives of the solution. In the special case k=2 we deduce that weak solutions in , , have locally H?lder continuous gradients. In the nondegenerate case we also show that weak solutions in , , have locally bounded second derivatives. Received February 25, 1999 / Accepted June 11, 1999 / Published online April 6, 2000     

Self-similar solutions for the anisotropic affine curve shortening problem

Similarity between the roles of the group on the equation for self-similar solutions of the anisotropic affine curve shortening problem and of the conformal group of on the Nirenberg problem for prescribed scalar curvature is explored. Sufficient conditions for the existence of affine self-similar curves are established. Received June 26, 1999 / Accepted January 28, 2000 / Published online December 8, 2000     

Nonlinear elliptic equations with natural growth in general domains

We prove the existence of solutions of nonlinear elliptic equations with first-order terms having “natural growth” with respect to the gradient. The assumptions on the source terms lead to the existence of possibly unbounded solutions (though with exponential integrability). The domain Ω is allowed to have infinite Lebesgue measure. Received: April 13, 2001; in final form: September 29, 2001?Published online: July 9, 2002     

Shooting with degree theory: Analysis of some weighted poly-harmonic systems

In this paper, the author establishes the existence of positive entire solutions to a general class of semilinear poly-harmonic systems, which includes equations and systems of the weighted Hardy–Littlewood–Sobolev type. The novel method used implements the classical shooting method enhanced by topological degree theory. The key steps of the method are to first construct a target map which aims the shooting method and the non-degeneracy conditions guarantee the continuity of this map. With the continuity of the target map, a topological argument is used to show the existence of zeros of the target map. The existence of zeros of the map along with a non-existence theorem for the corresponding Navier boundary value problem imply the existence of positive solutions for the class of poly-harmonic systems.     

Bounded traveling waves of the Burgers-Huxley equation

In order to investigate bounded traveling waves of the Burgers-Huxley equation, bifurcations of codimension 1 and 2 are discussed for its traveling wave system. By reduction to center manifolds and normal forms we give conditions for the appearance of homoclinic solutions, heteroclinic solutions and periodic solutions, which correspondingly give conditions of existence for solitary waves, kink waves and periodic waves, three basic types of bounded traveling waves. Furthermore, their evolutions are discussed to investigate the existence of other types of bounded traveling waves, such as the oscillatory traveling waves corresponding to connections between an equilibrium and a periodic orbit and the oscillatory kink waves corresponding to connections of saddle-focus.     

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.     

The Navier-Stokes equations in the weak- LnL^n space with time-dependent external force

Abstract. We consider the Navier-Stokes equations with time-dependent external force, either in the whole time or in positive time with initial data, with domain either the whole space, the half space or an exterior domain of dimension . We give conditions on the external force sufficient for the unique existence of small solutions in the weak- space bounded for all time. In particular, this result gives sufficient conditions for the unique existence and the stability of small time-periodic solutions or almost periodic solutions. This result generalizes the previous result on the unique existence and the stability of small stationary solutions in the weak- space with time-independent external force. Received: 30 March 1999 / Accepted: 21 September 1999 / Published online: 28 June 2000     

Existence of weak solutions for steady viscous fluids with general slip boundary conditions

We study the existence of weak solutions for stationary viscous fluids with general slip boundary conditions in this paper. Applying monotone operator theory, we first establish the existence result of weak solutions for an approximation problem. Then using the compactness methods and the point-wise convergence property of velocity gradients, we get the desired results.