Positive Solutions For p-Laplacian Equations Research Articles

Nonexistence of Global Positive Solutions for p-Laplacian Equations with Non-Linear Memory

The Cauchy problem in Rd,d≥1, for a non-local in time p-Laplacian equations is considered. The nonexistence of nontrivial global weak solutions by using the test function method is obtained.

Multiple positive solutions for p-Laplacian equations with integral boundary conditions

In this paper, we consider a class of boundary value problems for p-Laplacian equation (Φp(u′))′+h(t)f(t,u(t),u′(t))=0 with integral boundary conditions u(0)−αu′(0)=∫01g1(s)u(s)ds,u(1)+βu′(1)=∫01g2(s)u(s)ds. By using the Avery–Peterson fixed point theorem, we obtain the existence of at least three positive solutions.

On the existence threshold for positive solutions of p-Laplacian equations with a concave–convex nonlinearity

We study the following boundary value problem with a concave–convex nonlinearity: [Formula: see text] Here Ω ⊂ ℝnis a bounded domain and 1 < q < p < r < p*. It is well known that there exists a number Λq, r> 0 such that the problem admits at least two positive solutions for 0 < Λ < Λq, r, at least one positive solution for Λ = Λq, r, and no positive solution for Λ > Λq, r. We show that [Formula: see text] where λ1(p) is the first eigenvalue of the p-Laplacian. It is worth noticing that λ1(p) is the threshold for existence/nonexistence of positive solutions to the above problem in the limit case q = p.

Two positive solutions for p-Laplacian equations with convex and concave non-linearities and a non-smooth potential

We consider a non-linear elliptic problem driven by the p-Laplacian and a non-smooth potential (hemivariational inequality). We assume that the non-linearity involves the combined effects of both concave (i.e. p-sublinear) and convex (i.e. p-superlinear) terms. We show the existence of two positive solutions, using non-smooth critical point theory and the Ekeland variational principle.