Narrow Results

In this paper we study the questions of existence and uniqueness of solutions for equations of type -div a(x,Du) + γ(u) ∋ ϕ, posed in an open bounded subset Ω of ℝ^N, with nonlinear boundary conditions of the form a(x,Du) · η + β(u) ∋ ψ. The nonlinear elliptic operator div a(x,Du) modeled on the p-Laplacian operator Δ_p(u) = div(|Du|^p-2Du), with p > 1, γ and β maximal monotone graphs in ℝ² such that 0 ∈ γ(0) ∩ β(0), and the data ϕ ∈ L¹(Ω) and ψ ∈ L¹(∂ Ω). Since D(γ) ≠ ℝ, we are dealing with obstacle problems. For this kind of problems the existence of weak solution, in the usual sense, fails to be true for nonhomogeneous boundary conditions, so a new concept of solution has to be introduced.

The aim of this paper is investigating the multiplicity of weak solutions of the quasilinear elliptic equation $-{\mathrm{\Delta }}_{p}u+V(x)|u{|}^{p-2}u=g(x,u)$ in ${ℝ}^N$, where $1<p<+\infty$, the nonlinearity $g$ behaves as $|u{|}^{p-2}u$ at infinity and $V$ is a potential satisfying suitable assumptions so that an embedding theorem for weighted Sobolev spaces holds. Both the non-resonant and resonant cases are analyzed.

In this paper, we give necessary and sufficient conditions for existence of bounded and positive solutions of a nonlinear elliptic system arising from potential type problems.

In this paper, by using variational methods, we study the following elliptic problem

This paper studies the existence of positive solutions of the Dirichlet problem for the nonlinear equation involving p-Laplacian operator: -Δ_pu=λf(u) on a bounded smooth domain Ω in ℝⁿ. The authors extend part of the Crandall-Rabinowitz bifurcation theory to this problem. Typical examples are checked in detail and multiplicity of the solutions are illustrated. Then the stability for the associated parabolic equation is considered and a Fujita-type result is presented.

The authors show the regularity of weak solutions for some typical quasi-linear elliptic systems governed by two p-Laplacian operators. The weak solutions of the following problem with lack of compactness are proved to be regular when a(x) and α,β,p,q satisfy some conditions:

The aim of this paper is to prove the existence of multiple positive solutions to the following critical p–Laplacian system with singular potential

This work aims to study the existence and the regularity of positive solutions to a p-Laplacian system with nonlinearities of growth conditions. We focus on positive ground-state solutions and we assume that the nonlinearities are controlled by general polynomial functions, and we use a variational method to apply the mountain pass theorem which guarantees the existence of a super-solution in the sense of Hernandez, then we construct some compact operator T and some invariant set K where we can use the Leray–Schauder fixed point theorem. By the end of this paper, we establish an -estimation which allows to derive a property of regularity for such positive solutions.

We prove the existence of infinitely many singular radial positive solutions for a quasilinear elliptic system with no variational structure

We prove that for any λ ∈ IR, there is a sequence of eigencurves (µ_k(λ))_k for the nonlinear coupled elliptic system

In this paper, we study the existence for two positive solutions to a nonhomogeneous elliptic equation of fourth order with a parameter λ such that . The first solution has a negative energy while the energy of the second one is positive for 0 < λ < λ₀ and negative for . The values λ₀ and are given under variational form and we show that every corresponding critical point is solution of the nonlinear elliptic problem (with a suitable multiplicative term).