In this paper we give some existence and nonexistence results of non trivial solutions of nonlinear elliptic systems involving the p-Laplacian.

The main goal in this paper is to prove the existence of radial positive solutions of the quasilinear elliptic system⎧ -Δpu = f(x,u,v) in Ω,⎨ -Δqv = g(x,u,v) in Ω,⎩ u = v = 0 on ∂Ω,where Ω is a ball in RN and f, g are positive continuous functions satisfying f(x, 0, 0) = g(x, 0, 0) = 0 and some growth conditions which correspond, roughly speaking, to superlinear problems. Two different sets of conditions, called strongly and weakly coupled, are given in order to obtain existence. We use...

We study the existence and nonexistence in the large of radial solutions to a parabolic-elliptic system with natural (no-flux) boundary conditions describing the gravitational interaction of particles. The blow-up of solutions defined in the n-dimensional ball with large initial data is connected with the nonexistence of radial stationary solutions with a large mass.

We study the existence of stationary and evolution solutions to a parabolic-elliptic system with natural (no-flux) boundary conditions describing the gravitational interaction of particles.

By a sub-super solution argument, we study the existence of positive solutions for the system ⎧$-{\Delta }_{p}u=a₁\left(x\right)F₁(x,u,v)$ in Ω, ⎪$-{\Delta }_{q}v=a₂\left(x\right)F₂(x,u,v)$ in Ω, ⎨u,v > 0 in Ω, ⎩u = v = 0 on ∂Ω, where Ω is a bounded domain in ${ℝ}^N$ with smooth boundary or $\Omega ={ℝ}^N$. A nonexistence result is obtained for radially symmetric solutions.

We consider the existence and nonexistence of solutions for the following singular quasi-linear elliptic problem with concave and convex nonlinearities: ⎧ $-{div\left(\right|x|}^{-ap}{\left|\nabla u\right|}^{p-2}{\nabla u)+h\left(x\right)|u|}^{p-2}u=g\left(x\right){\left|u\right|}^{r-2}u$, x ∈ Ω, ⎨ ⎩ ${\left|x\right|}^{-ap}{\left|\nabla u\right|}^{p-2}\partial u/\partial \nu =\lambda f\left(x\right){\left|u\right|}^{q-2}u$, x ∈ ∂Ω, where Ω is an exterior domain in ${ℝ}^N$, that is, $\Omega ={ℝ}^N\setminus D$, where D is a bounded domain in ${ℝ}^N$ with smooth boundary ∂D(=∂Ω), and 0 ∈ Ω. Here λ > 0, 0 ≤ a < (N-p)/p, 1 < p< N, ∂/∂ν is the outward normal derivative on ∂Ω. By the variational method, we prove the existence of multiple solutions. By the test function method,...

In this work we study the problem $u_t{-div\left(\right|x|}^{-2\gamma }\nabla u)=\lambda \frac{u^{\alpha }}{{\left|x\right|}^{2(\gamma +1)}}+f$ in $\Omega \times (0,T)$, $u\ge 0$ in $\Omega \times (0,T)$, $u=0$ on $\partial \Omega \times (0,T)$, $u(x,0)=u_0\left(x\right)$ in $\Omega$, $\Omega \subset {ℝ}^N$$(N\ge 2)$ is a bounded regular domain such that $0\in \Omega$, $\lambda >0$, $\alpha >0$, $-\infty <\gamma <\left(N-2\right)/2$, $f$ and $u_0$ are positive functions such...

Problems of existence and nonexistence of global nontrivial solutions to quasilinear evolution differential inequalities in a product of cones are investigated. The proofs of the nonexistence results are based on the test-function method developed, for the case of the whole space, by Mitidieri, Pohozaev, Tesei and Véron. The existence result is established using the method of supersolutions.

It is known that degenerate parabolic equations exhibit somehow different phenomena when we compare them with their elliptic counterparts. Thus, the problem of existence and properties of the Green function for degenerate parabolic boundary value problems is not completely solved, even after the contributions of [GN] and [GW4], in the sense that the existence problem is still open, even if the a priori estimates proved in [GN] will be crucial in our approach (...).