Existence and multiplicity of solutions for Neumann problems

In this paper we examine semilinear and nonlinear Neumann problems with a nonsmooth locally Lipschitz potential function. Using variational methods based on the nonsmooth critical point theory, for the semilinear problem we prove a multiplicity result under conditions of double resonance at higher eigenvalues. Our proof involves a nonsmooth extension of the reduction method due to Castro-Lazer-Thews. The nonlinear problem is driven by the p-Laplacian. So first we make some observations about the beginning of the spectrum of (−Δp,W1,p(Z)). Then we prove an existence and multiplicity result. The existence result permits complete double resonance. The multiplicity result specialized in the semilinear case (i.e. p=2) corresponds to the super-sub quadratic situation.     

Uniqueness of solutions for some nonlinear Dirichlet problems

We consider here a class of nonlinear Dirichlet problems, in a bounded domain , of the form

Uniqueness and structure of solutions to the Dirichlet problem for an elliptic system

In this paper, we consider the Dirichlet problem for an elliptic system on a ball in R². By investigating the properties for the corresponding linearized equations of solutions, and adopting the Pohozaev identity and Implicit Function Theorem, we show the uniqueness and the structure of solutions.     

Concentration and multiplicity of solutions for a fourth-order equation with critical nonlinearity

In this paper, we consider the problem (_Pε)$(P_{\epsilon })$ : Δ²u=u^n+4/n-4+εu,u>0${\mathrm{\Delta }}^{2}u=u^{n+4/n-4}+\epsilon u,u>0$ in Ω,u=Δu=0$\Omega ,u=\mathrm{\Delta }u=0$ on ∂Ω$\mathrm{\partial }\Omega$, where Ω$\Omega$ is a bounded and smooth domain in Rⁿ,n>8${\mathbb{R}}^n,n>8$ and ε>0$\epsilon >0$. We analyze the asymptotic behavior of solutions of (_Pε)$(P_{\epsilon })$ which are minimizing for the Sobolev inequality as ε→0$\epsilon \to 0$ and we prove existence of solutions to (_Pε)$(P_{\epsilon })$ which blow up and concentrate around a critical point of the Robin's function. Finally, we show that for ε$\epsilon$ small, (_Pε)$(P_{\epsilon })$ has at least as many solutions as the Ljusternik–Schnirelman category of Ω$\Omega$.     

Existence of solutions for a class of singular quasilinear elliptic resonance problems

The paper concerns a resonance problem for a class of singular quasilinear elliptic equations in weighted Sobolev spaces. The equation set studied is one of the most useful sets of Navier-Stokes equations; these describe the motion of viscous fluid substances such as liquids, gases and so on. By using Galerkin-type techniques, the Brouwer fixed point theorem, and a new weighted compact Sobolev-type embedding theorem established by Shapiro, we show the existence of a nontrivial solution.     

Existence and non-existence of solutions for a class of Monge-Ampère equations

We study the boundary value problems for Monge-Ampère equations: detD²u=e−u in Ω⊂Rn, n?1, u_|∂Ω=0. First we prove that any solution on the ball is radially symmetric by the argument of moving plane. Then we show there exists a critical radius such that if the radius of a ball is smaller than this critical value there exists a solution, and vice versa. Using the comparison between domains we can prove that this phenomenon occurs for every domain. Finally we consider an equivalent problem with a parameter detD²u=e−tu in Ω, u_|∂Ω=0, t?0. By using Lyapunov-Schmidt reduction method we get the local structure of the solutions near a degenerate point; by Leray-Schauder degree theory, a priori estimate and bifurcation theory we get the global structure.     

Multiple solutions for a class of elliptic equations with jumping nonlinearities

We consider a semilinear elliptic Dirichlet problem with jumping nonlinearity and, using variational methods, we show that the number of solutions tends to infinity as the number of jumped eigenvalues tends to infinity. In order to prove this fact, for every positive integer k we prove that, when a parameter is large enough, there exists a solution which presents k interior peaks. We also describe the asymptotic behaviour and the profile of this solution as the parameter tends to infinity.     

On a new Kato class and positive solutions of Dirichlet problems for the fractional Laplacian in bounded domains

The purpose of this paper is to extend some results of the potential theory of an elliptic operator to the fractional Laplacian (−Δ)^α/2, 0<α<2, in a bounded C^1,1 domain D in Rⁿ. In particular, we introduce a new Kato class K_α(D) and we exploit the properties of this class to study the existence of positive solutions of some Dirichlet problems for the fractional Laplacian.     

Stability results for Mountain Pass and Linking type solutions of semilinear problems involving Dirichlet forms

Some stability results for Mountain Pass and Linking type solutions of semilinear problems involving a very general class of Dirichlet forms are stated. The non linear terms are supposed to have a suitable superlinear growth and the family of Dirichlet forms is required to be dominated from below and from above by a fixed diffusion type form. Some concrete examples are also given.     

Lower and upper solutions for elliptic problems in nonsmooth domains

In this paper we prove some existence results of semilinear Dirichlet problems in nonsmooth domains in presence of lower and upper solutions well-ordered or not. We first prove existence results in an abstract setting using degree theory. We secondly apply them for domains with conical points.     

The continuous dependence on parameters of solutions for a class of elliptic problems on exterior domains

The goal of this paper is to discuss the continuous dependence of solutions on functional parameters for the following semilinear elliptic partial differential equation: , for x∈Ω_r0?{x∈Rⁿ,n≥3,‖x‖>r₀} and v∈V, where V stands for some functional space. Our approach covers the case when f may change sign and admits general growth. As an additional result, the characterization of the radius r₀ for which our problem possesses at least one positive evanescent solution in the exterior domain Ω_r0 is described and numerically illustrated. Our approach relies on the subsolution and supersolution method and on a lemma due to Noussair and Swanson.     

Existence and uniqueness of positive solutions for singular nonlinear elliptic boundary value problems

In this paper we establish the existence and the uniqueness of positive solutions for Dirichlet boundary value problems of nonlinear elliptic equations with singularity. We obtain the existence and the uniqueness by using the mixed monotone method in the cone theory. Moreover, we give an iterative method of constructing the solution. The rate of convergence of the iterative sequence is analyzed.     

An existence theorem for weak solutions for a class of elliptic partial differential systems in Orlicz spaces

We prove the existence of a weak solution of the Dirichlet problem for a class of elliptic partial differential systems in separable Orlicz–Sobolev spaces.     

Stability and multiplicity of solutions for the thermistor problem

The stability of the stationary solution of the thermistor as a circuit element is studied using a Liapunov functional and the Hale–LaSalle invariance principle. The asymptotic stability of a class of periodic solutions is also considered. Received: November 22, 1999; in final form: May 23, 2001?Published online: May 29, 2002     

Existence and multiplicity of nonnegative solutions for semilinear elliptic problems involving nonlinearities indefinite in sign

Let r,s∈]1,2[$r,s\in ]1,2[$ and λ,μ∈]0,+∞[$\lambda ,\mu \in ]0,+\infty [$. In this paper, we deal with the existence and multiplicity of nonnegative and nonzero solutions of the Dirichlet problem with 0$0$ boundary data for the semilinear elliptic equation −Δu=λu^s−1−u^r−1$-\Delta u=\lambda u^{s-1}-u^{r-1}$ in Ω⊂R^N$\Omega \subset {\mathbb{R}}^N$, where N≥2$N\ge 2$. We prove that there exists a positive constant Λ$\Lambda$ such that the above problem has at least two solutions, at least one solution or no solution according to whether λ>Λ$\lambda >\Lambda$, λ=Λ$\lambda =\Lambda$ or λ<Λ$\lambda <\Lambda$. In particular, a result by Hernandéz, Macebo and Vega is improved and, for the semilinear case, a result by Díaz and Hernandéz is partially extended to higher dimensions. Finally, an answer to a conjecture, recently stated by the author, is also given.