Advanced Search

In this note, the existence of non-negative solutions for some multivalued non-positone elliptic problems is studied.

In this paper we deal with the existence of critical points for functionals defined on the Sobolev space $W_0^{1,2}\left(\mathrm{\Omega }\right)$ by $J\left(v\right)={\int }_{\mathrm{\Omega }}\mathfrak{I}\left(x,v,Dv\right)dx$, $v\in W_0^{1,2}\left(\mathrm{\Omega }\right)$, where $\mathrm{\Omega }$ is a bounded, open subset of ${\mathbb{R}}^N$. Since the differentiability can fail even for very simple examples of functionals defined through multiple integrals of Calculus of Variations, we give a suitable version of the Ambrosetti-Rabinowitz Mountain Pass Theorem, which enables us to the study of critical points for functionals which are not differentiable in all directions. Then we...

We study the Ambrosetti–Prodi and Ambrosetti–Rabinowitz problems.We prove for the first one the existence of a continuum of solutions with shape of a reflected $C$ ($\supset$-shape). Next, we show that there is a relationship between these two problems.

We deal with the boundary value problem $$\begin{array}{ccccccc}\hfill -\Delta u\left(x\right)& ={\lambda }_{1}u\left(x\right)+g\left(\nabla u\left(x\right)\right)+h\left(x\right),\hfill & & x\in \Omega u\left(x\right)\hfill & \hfill =0,& & \hfill x\in \partial \Omega \end{array}$$ where $\Omega \subset {ℝ}^N$ is an smooth bounded domain, ${\lambda }_1$ is the first eigenvalue of the Laplace operator with homogeneous Dirichlet boundary conditions on $\Omega$, $h\in L^{max\{2,N/2\}}\left(\Omega \right)$ and $g:{ℝ}^N⟶ℝ$ is bounded and continuous. Bifurcation theory is used as the right framework to show the existence of solution provided that $g$ satisfies certain conditions on the origin and at infinity.

We study a comparison principle and uniqueness of positive solutions for the homogeneous Dirichlet boundary value problem associated to quasi-linear elliptic equations with lower order terms. A model example is given by $-\Delta u+\lambda \frac{{\left|\nabla u\right|}^2}{u^r}=f\left(x\right),\lambda ,r>0.$ The main feature of these equations consists in having a quadratic gradient term in which singularities are allowed. The arguments employed here also work to deal with equations having lack of ellipticity or some dependence...