Infinitely many solutions for a class of quasilinear elliptic equations with p-Laplacian in R^N (original) (raw)

Existence and nonexistence results for quasilinear elliptic equations involving the p-laplacian

The paper deals with the study of a quasilinear elliptic equation involving the p-laplacian with a Hardy-type singular potential and a critical nonlinearity. Existence and nonexistence results are first proved for the equation with a concave singular term. Then we study the critical case relate to Hardy inequality, providing a description of the behavior of radial solutions of the limiting problem and obtaining existence and multiplicity results for perturbed problems through variational and topological arguments.

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.

The Existence of Multiple Solutions to Quasilinear Elliptic Equations

Bulletin of the London Mathematical Society, 2005

Using Morse theory and the truncation technique, a proof is given of the existence of at least three nontrivial solutions for a class of p-Laplacian equations. When p = 2, the existence of four nontrivial solutions is also considered.

Multiplicity of Solutions for Elliptic Problems with Critical Exponent or with a Nonsymmetric Term

Transactions of the American Mathematical Society, 1991

We study the existence of solutions for the following nonlinear degenerate elliptic problems in a bounded domain Í2 c R-div(|VK|p~2VK) = \u\p"~2u + k\u\q~2u, A > 0, where p' is the critical Sobolev exponent, and u\sn = 0. By using critical point methods we obtain the existence of solutions in the following cases: If p < q < p* , there exists A0 > 0 such that for all A > A0 there exists a nontrivial solution. If max(p, p*-pKp-1)) < q < p* , there exists nontrivial solution for all A>0. If 1 < q < p there exists A, such that, for 0 < A < A, , there exist infinitely many solutions. Finally, we obtain a multiplicity result in a noncritical problem when the associated functional is not symmetric.

Critical points of solutions to quasilinear elliptic problems

Nonlinear Analysis: Theory, Methods & Applications, 2012

In this paper, we mainly investigate the critical points associated to solutions u of a quasilinear elliptic equation with nonhomogeneous Dirichlet boundary conditions in a connected domain in R 2. Based on the fine analysis about the distribution of connected components of a super-level set {x ∈ : u(x) > t} for any min ∂ u(x) < t < max ∂ u(x), we obtain the geometric structure of interior critical points of u. Precisely, when is simply connected, we develop a new method to prove k i=1 m i + 1 = N, where m 1 , • • • , m k are the respective multiplicities of interior critical points x 1 , • • • , x k of u and N is the number of global maximal points of u on ∂. When is an annular domain with the interior boundary γ I and the external boundary γ E , where u| γ I = H, u| γ E = ψ(x) and ψ(x) has N local (global) maximal points on γ E. For the case ψ(x) ≥ H or ψ(x) ≤ H or min γ E ψ(x) < H < max γ E ψ(x), we show that k i=1 m i ≤ N (either k i=1 m i = N or k i=1 m i + 1 = N).

Critical points of A-solutions of quasilinear elliptic equations

1999

Critical points of solutions to degenerate elliptic equations in R n , n ≥ 2, consist of good (N -points) and bad points. Pseudoharmonic functions (n = 2), in the sense of Morse, have good points only. We give an estimate for the modulus of continuity of a generalized solution at an N -point. An analog of Sard's theorem is proved.

Multiplicity of Solutions for Quasilinear Elliptic Problems

arXiv: Analysis of PDEs, 2017

It is established existence, uniqueness and multiplicity of solutions for a quasilinear elliptic problem problems driven by Phi\PhiPhi-Laplacian operator. Here we consider the reflexive and nonreflexive cases using an auxiliary problem. In order to prove our main results we employ variational methods, regularity results and truncation techniques.