Multiple positive solutions for a quasilinear elliptic equation with critical exponential nonlinearity

Let Ω be a bounded domain in R^N,N≥2, with C² boundary. In this work, we study the existence of multiple positive solutions of the following problem:

     

Multiple solutions for quasilinear elliptic equations with a finite potential well

We consider multiplicity of solutions for a class of quasilinear problems which has received considerable attention in the past, including the so called Modified Nonlinear Schrödinger Equations. By combining a new variational approach via q-Laplacian regularization and the compactness arguments from [4] we establish infinitely many bound state solutions for the quasilinear Schrödinger type equations, extending the earlier work of [4] for semilinear equations.     

Some existence results for quasilinear elliptic equations in a finite cylinder

We study the existence of solutionsu in the finite cylinderS _a=(?a, a)×ω of the quasilinear elliptic equation $$\sum\limits_{i,j = 1}^n {a_{i,j} (x)\partial _{i,j} u + f(x,u,\nabla u) = 0} $$ with Dirichlet boundary condition on flat parts of σS _a and Neumann condition on the curved parts. In this paper, we focus on the technicality caused by the “corners” ofS _a. We prove the existence of such solutions provided that suitable sub and super solutions are known and under the condition that the coefficientsa _1,i,i#1 vanish on the corners. We also prove a more general result in ?².     

Multiple solutions for a quasilinear second order differential equation depending on a parameter

The purpose of this paper is to use a very recent three critical points theorem due to Bonanno and Marano to establish the existence of at least three solutions for the quasilinear second order differential equation on a compact interval [a, b] ? R

$$\left( {\begin{array}{*{20}{c}}{t - 1} \\3 \end{array}} \right) {u\left( a \right)}$$

under appropriate hypotheses. We exhibit the existence of at least three (weak) solutions and, and the results are illustrated by examples.

     

Multiple symmetric results for singular quasilinear elliptic systems with critical homogeneous nonlinearity

This paper deals with the existence and multiplicity of symmetric solutions for a class of singular quasilinear elliptic systems with critical homogeneous nonlinearity in a bounded symmetric domain. Applying variational methods and the symmetric criticality principle of Palais, we establish several existence and multiplicity results of G‐symmetric solutions under some appropriate assumptions on the weighted functions and the parameters. Copyright © 2016 John Wiley & Sons, Ltd.     

Multibump solutions for quasilinear elliptic equations

The current paper is concerned with constructing multibump type solutions for a class of quasilinear Schrödinger type equations including the Modified Nonlinear Schrödinger Equations. Our results extend the existence results on multibump type solutions in Coti Zelati and Rabinowitz (1992) [17] to the quasilinear case. Our work provides a theoretic framework for dealing with quasilinear problems, which lack both smoothness and compactness, by using more refined variational techniques such as gluing techniques, Morse theory, Lyapunov–Schmidt reduction, etc.     

Multiple sign‐changing solutions for a class of quasilinear elliptic equations

In this paper, we study the following quasilinear Schrödinger equations: where Ω is a bounded smooth domain of , . Under some suitable conditions, we prove that this equation has three solutions of mountain pass type: one positive, one negative, and sign‐changing. Furthermore, if g is odd with respect to its second variable, this problem has infinitely many sign‐changing solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

Multiple existence of non-radial solutions with group invariance for sublinear elliptic equations

We deal with sublinear elliptic equations in a ball and prove the existence of infinitely many solutions which are not radially symmetric but G invariant. Here G is any closed subgroup of the orthogonal group and is not transitive on the unit sphere.     

Entire solutions of quasilinear elliptic equations

We study entire solutions of non-homogeneous quasilinear elliptic equations, with Eqs. (1) and (2) below being typical. A particular special case of interest is the following: Let u be an entire distribution solution of the equation Δpu=|u|^q−1u, where p>1. If q>p−1 then u≡0. On the other hand, if 0<q<p−1 and u(x)=o(|x|^{p/(p−q−1)}) as |x|→∞, then again u≡0. If q=p−1 then u≡0 for all solutions with at most algebraic growth at infinity.     

Existence of solutions for degenerate quasilinear elliptic equations

This paper deals with a class of degenerate quasilinear elliptic equations of the form −div(a(x,u,∇u)=g−div(f), where a(x,u,∇u) is allowed to be degenerate with the unknown u. We prove existence of bounded solutions under some hypothesis on f and g. Moreover we prove that there exists a renormalized solution in the case where g∈L¹(Ω) and f∈(L^p′(Ω))^N.     

Multiple positive solutions for a critical quasilinear elliptic system

The existence of cat_Ω(Ω) positive solutions for the p-Laplacian system with convex and Sobolev critical nonlinearities is obtained by some standard variational methods, whose key is to construct homotopies between Ω and levels of the functional J_λ,μ, and some analytical techniques.