Multiplicity of weak solutions for a class of nonuniformly elliptic equations of p -Laplacian type (original) (raw)
Related papers
Some Remarks on a Class of Nonuniformly Elliptic Equations of p-Laplacian Type
Acta Applicandae Mathematicae, 2008
This paper deals with the existence of weak solutions in W 1 0 () to a class of elliptic problems of the form − div(a(x, ∇u)) = λ 1 |u| p−2 u + g (u) − h in a bounded domain of R N. Here a satisfies |a (x, ξ)| c 0 h 0 (x) + h 1 (x) |ξ | p−1 for all ξ ∈ R N , a.e. x ∈ , h 0 ∈ L p p−1 (), h 1 ∈ L 1 loc (), h 1 (x) 1 for a.e. x in ; λ 1 is the first eigenvalue for − p on with zero Dirichlet boundary condition and g, h satisfy some suitable conditions.
Complex Variables and Elliptic Equations, 2011
We discuss the problem -div(a(x, ru)) ¼ m(x)juj r(x)À2 u þ n(x)juj s(x)À2 u in , where is a bounded domain with smooth boundary in R N (N ! 2), and div(a(x, ru)) is a p(x)-Laplace type operator with 1 < r(x) < p(x) < s(x). We show the existence of infinitely many nontrivial weak solutions in W 1,pðxÞ 0 ð Þ. Our approach relies on the theory of the variable exponent Lebesgue and Sobolev spaces combined with adequate variational methods and a variation of the Mountain Pass lemma and critical point theory.
Multiplicity results for nonlinear elliptic equations
Let Ω be a bounded domain in R N , N ≥ 3, and p = 2N N −2 the limiting Sobolev exponent. We show that for f ∈ H 1 0 (Ω) * , satisfying suitable conditions, the nonlinear elliptic problem −∆u = |u| p−2 u + f in Ω u = 0 on ∂Ω has at least three solutions in H 1 0 (Ω).
Multiple Solutions for p-Laplacian Type Problems with an Asymptotically p-linear Term
Analysis and Topology in Nonlinear Differential Equations, 2014
The aim of this paper is investigating the existence of weak solutions of the quasilinear elliptic model problem − div(A(x, u) |∇u| p−2 ∇u) + 1 p At(x, u) |∇u| p = f (x, u) in Ω, u = 0 on ∂Ω, where Ω ⊂ R N is a bounded domain, N ≥ 2, p > 1, A is a given function which admits partial derivative At(x, t) = ∂A ∂t (x, t) and f is asymptotically p-linear at infinity. Under suitable hypotheses both at the origin and at infinity, and if A(x, •) is even while f (x, •) is odd, by using variational tools, a cohomological index theory and a related pseudo-index argument, we prove a multiplicity result if p > N in the non-resonant case.
Multiplicity results for some quasilinear elliptic problems
Topological Methods in Nonlinear Analysis, 2009
In this paper, we study multiplicity of weak solutions for the following class of quasilinear elliptic problems of the form where Ω is a bounded domain in R n with smooth boundary ∂Ω, 1 < q < 2 < p ≤ n, λ is a real parameter, ∆pu = div(|∇u| p-2 ∇u) is the p-Laplacian and the nonlinearity g(u) has subcritical growth. The proofs of our results rely on some linking theorems and critical groups estimates.
Nonuniformly Elliptic Equations of P-Laplacian Type
Nonlinear Analysis, 2005
This paper deals with the existence of a generalized solution in W 1,p 0 ( ) to a nonuniformly nonlinear elliptic equation of the form −div(a(x, ∇u)) = f (x, u) in a bounded domain of R n . Here