We consider the Neumann problem involving the critical Sobolev exponent and a nonhomogeneous boundary condition. We establish the existence of two solutions. We use the method of sub- and supersolutions, a local minimization and the mountain-pass principle.

We investigate the effect of the topology of the boundary ∂Ω and of the graph topology of the coefficient Q on the number of solutions of the nonlinear Neumann problem $\left(1_d\right)$.

We consider the following singularly perturbed elliptic problem$$\begin{array}{cc}\hfill {ϵ}^2\Delta u-u+f\left(u\right)& =0,u\>0\text{in}\Omega ,\hfill \\ \hfill ϵ\frac{\partial u}{\partial \nu }+\lambda u& =0\text{on}\partial \Omega ,\hfill \end{array}$$where $f$ satisfies some growth conditions, $0\le \lambda \le +\infty$, and $\Omega \subset {ℝ}^N$ ($N\>1$) is a smooth and bounded domain. The cases $\lambda =0$ (Neumann problem) and $\lambda =+\infty$ (Dirichlet problem) have been studied by many authors in recent years. We show that, there exists a generic constant ${\lambda }_{*}\>1$ such that, as $ϵ\to 0$, the least energy solution has a spike near the boundary if $\lambda \le {\lambda }_{*}$, and has an interior spike near the innermost part of the domain if $\lambda \>{\lambda }_{*}$. Central to our study is the corresponding problem...

We establish two existence results for elliptic boundary-value problems with discontinuous nonlinearities. One of them concerns implicit elliptic equations of the form ψ(-Δu) = f(x,u). We emphasize that our assumptions permit the nonlinear term f to be discontinuous with respect to the second variable at each point.

Boundary value problems for second order linear elliptic equations with coefficients having discontinuities of the first kind on an infinite number of smooth surfaces are studied. Existence, uniqueness and regularity results are furnished for the diffraction problem in such a bounded domain, and for the corresponding transmission problem in all of ${ℝ}^N$. The transmission problem corresponding to the scattering of acoustic plane waves by an infinitely stratified scatterer, consisting of layers with physically...

In this review article we present an overview on some a priori estimates in $L^p$, $p>1$, recently obtained in the framework of the study of a certain kind of Dirichlet problem in unbounded domains. More precisely, we consider a linear uniformly elliptic second order differential operator in divergence form with bounded leading coeffcients and with lower order terms coefficients belonging to certain Morrey type spaces. Under suitable assumptions on the data, we first show two $L^p$-bounds, $p>2$, for the solution...

Let $\Omega$ be a smooth bounded domain in ${ℝ}^N,N\>1$ and let $n\in {\mathbb{N}}^{*}$. We prove here the existence of nonnegative solutions $u_n$ in $BV\left(\Omega \right)$, to the problem$$\left(P_n\right)\left\{\begin{array}{cc}-div\sigma +2n\left({\int }_{\Omega }u-1\right){sign}^{+}\left(u\right)=0\hfill & \text{in}\Omega ,\hfill \\ \sigma ·\nabla u=\left|\nabla u\right|\hfill & \text{in}\Omega ,\hfill \\ u\text{is}\text{not}\text{identically}\text{zero},-\sigma ·\overrightarrow{n}u=u\hfill & \text{on}\partial \Omega ,\hfill \end{array}\right.$$where $\overrightarrow{n}$ denotes the unit outer normal to $\partial \Omega$, and ${\mathrm{sign}}^{+}\left(u\right)$ denotes some $L^{\infty }\left(\Omega \right)$ function defined as:$${\mathrm{sign}}^{+}\left(u\right).u=u^{+},0\le {\mathrm{sign}}^{+}\left(u\right)\le 1.$$Moreover, we prove the tight convergence of $u_n$ towards one of the first eingenfunctions for the first $1-$Laplacian Operator $-{\Delta }_1$ on $\Omega$ when $n$ goes to $+\infty$.