Multiplicity of solutions for quasilinear elliptic systems with critical growth

We study the existence of multiple solutions for a quasilinear elliptic system of gradient type with critical growth and the possibility of coupling on the subcritical term. The solutions are obtained from a version of the Symmetric Mountain Pass Theorem. The Concentration-Compactness Principle allows to verify that the Palais-Smale condition is satisfied below a certain level. The authors were partially supported by CNPq/Brazil     

Multiple solutions for elliptic systems with nonlinearities of arbitrary growth

We prove the existence of infinitely many solutions for symmetric elliptic systems with nonlinearities of arbitrary growth. Moreover, if the symmetry of the problem is broken by a small enough perturbation term, we find at least three solutions. The proofs utilise a variational setting given by de Figueiredo and Ruf in order to prove an existence's result and the “algebraic” approach based on the Pohozaev's fibering method.     

Sign-changing solutions of critical growth nonlinear elliptic systems

In this paper, we are concerned with the existence of sign-changing solutions of a class of nonlinear elliptic systems with critical growth.     

Existence of positive solutions to semilinear elliptic systems with supercritical growth

We establish the existence of positive entire solutions to cooperative systems of semilinear elliptic equations involving nonlinearities with critical and supercritical growth. Consequently, we obtain existence results to several well-known model examples such as systems of the Hénon, Lane–Emden, and stationary Schrödinger types. The main technique for generating our results relies on a topological approach for the shooting method combined with nonexistence results to closely related boundary value problems.     

Weak Harnack's Inequality for non-negative solutions of elliptic equations with potential

We present an alternative and shorter proof to a weak Harnack inequality for non-negative solutions of divergence structure elliptic equations with potentials from the Kato class.

     

Weak solutions for a high-order pseudo-parabolic equation with variable exponents

In this paper, the authors investigate the existence and uniqueness of weak solutions of the initial and boundary value problem for a fourth-order pseudo-parabolic equation with variable exponents of non-linearity. Finally, the authors also obtain a long-time behaviour of weak solutions.     

Nonnegative solutions for nonlinear elliptic systems

In this paper, we show that the semilinear elliptic systems of the form

(0.1)     

Existence of solutions of sublinear elliptic systems with quadratic growth in the gradient

In this paper, we deal with the following sublinear elliptic system: $${{\left\{\begin{array}{ll}-\Delta u + u + |\nabla u|^2 = a(x)|v|^p+ f,\quad x \in \mathbb{R}^N,\\ -\Delta v + v + |\nabla v|^2 = b(x)|u|^q+ g,\quad x \in \mathbb{R}^N,\\\end{array}\right.}}$$ where 0 <  p <  1 and 0 <  q <  1. Under suitable assumptions on the terms a, b, f and g and by using the Schauder fixed point theorem, we obtain a solution for an approximated system. The limit of the approximated solutions is a nonnegative solution.     

Entropy solutions for a doubly nonlinear elliptic problem with variable exponent

We study the boundary value problem in Ω, u=0 on ∂Ω, where Ω is a smooth bounded domain in RN (N?3) and is a p(x)-Laplace type operator with p(.):Ω→[1,+∞) a measurable function and b a continuous and nondecreasing function from R→R. We prove the existence and uniqueness of an entropy solution for L¹-data f.     

Infinitely many solutions for elliptic systems with critical exponents

One class of critical growth elliptic systems of two equations is considered on a bounded domain. By using fountain theorem and concentration estimates, we establish the existence of infinitely many solutions in higher values of dimension N?7.