Solutions For Nonhomogeneous Elliptic Equations Research Articles

On nonhomogeneous elliptic equations with the Hardy—Leray potentials

In this paper, we present some suitable distributional identities of solutions for nonhomogeneous elliptic equations involving the Hardy—Leray potentials and study qualitative properties of the solutions to the corresponding nonhomogeneous problems in this distributional sense. We address some applications on the nonexistence of some nonhomogeneous problems with the Hardy—Leray potentials and the nonexistence principle eigenvalue with some indefinite potentials.

Positive maximal and minimal solutions for non-homogeneous elliptic equations depending on the gradient

We are concerned with positive maximal and minimal solutions for non-homogeneous elliptic equations of the form−div(a(|∇u|p)|∇u|p−2∇u)=f(x,u,∇u) in Ω, supplied with Dirichlet boundary conditions. First we localize maximal and minimal solutions between not necessarily bounded sub-super solutions. Then using a uniform gradient estimate, which seems of independent interest, we show the existence of positive maximal and minimal solutions in some situations. More precisely, we obtain positive maximal and minimal solution to some classes of non-homogeneous equations depending on the gradient which may be perturbed by unbounded, singular or logistic sources.

Multiple Solutions for Nonhomogeneous Elliptic Equations Involving Critical Caffarelli–Kohn–Nirenberg Exponent

In this paper, we consider a nonhomogeneous singular elliptic equation involving a critical Caffarelli–Kohn–Nirenberg exponent. Using Ekeland’s Variational Principle, the Mountain Pass Lemma and the Nehari manifold, we establish the existence of at least two solutions.

Positive solutions for nonhomogeneous elliptic equations involving critical Sobolev exponent

In this paper, we study the existence and multiplicity of positive solutions for the Dirichlet problem − Δ u = λ f ( z ) | u | q − 2 u + g ( z ) | u | 2 ∗ − 2 u in Ω , where λ > 0 , 1 ≤ q < 2 , 2 ∗ = 2 N / ( N − 2 ) , 0 ∈ Ω ⊂ R N ( N ≥ 3 ) is a bounded domain with smooth boundary ∂ Ω , and f , g are nonnegative continuous functions on Ω ¯ .

Multiple positive solutions of non-homogeneous elliptic equations in exterior domains

In this paper, we consider the following non-homogeneous elliptic problem\begin{gather*} -\Delta u+u=\lambda(u^{p-1}+h(x))\text{ in }\varOmega, \\ u&gt;0\text{ in }\varOmega,\quad u\in H_0^{1}(\varOmega), \end{gather*}where .

Multiple Positive Solutions of Nonhomogeneous Elliptic Equations in Unbounded Domains

We will show that under suitable conditions onfandh, there exists a positive numberλ∗such that the nonhomogeneous elliptic equation−Δu+u=λ(f(x,u)+h(x))inΩ,u∈H01(Ω),N≥2, has at least two positive solutions ifλ∈(0,λ∗), a unique positive solution ifλ=λ∗, and no positive solution ifλ&gt;λ∗, whereΩis the entire space or an exterior domain or an unbounded cylinder domain or the complement in a strip domain of a bounded domain. We also obtain some properties of the set of solutions.

Multiplicity and concentration of solutions for nonhomogeneous elliptic equations in multi-bump domains

In this paper, we study the decomposition of the filtration of the Nehari manifold via the variation of domain shape. Furthermore, we use this result to prove that the nonhomogeneous elliptic equations in an m -bump domain have at least m + 1 positive solutions.

Multiple positive solutions for nonhomogeneous elliptic equations

Multiple positive solutions for nonhomogeneous elliptic equations

Existence of entire positive solutions for nonhomogeneous elliptic equations

Existence of entire positive solutions for nonhomogeneous elliptic equations

On the existence of multiple solutions of nonhomogeneous elliptic equations involving critical Sobolev exponents

Letp=2N/(N −2),N ≥ 3 be the limiting Sobolev exponent and Ω e ℝ N a bounded smooth domain. We show that for ħ eH −1(Ω),f satisfies some conditions then−Δu=c 1 u p−1 +f(x,u) + ħ admits at least two positive solutions.

Multiple solutions for nonhomogeneous elliptic equations involving critical Sobolev exponent

We consider the following problem:where is continuous on RN and h(x)≢0. By using Ekeland's variational principle and the Mountain Pass Theorem without (PS) conditions, through a careful inspection of the energy balance for the approximated solutions, we show that the probelm (*) has at least two solutions for some λ* &gt; 0 and λ ∈ (0, λ*). In particular, if p = 2, in a different way we prove that problem (*) with λ ≡ 1 and h(x) ≧ 0 has at least two positive solutions as