On positive solution for a class of degenerate quasilinear elliptic positone/semipositone systems

This paper deals with the existence and nonexistence of positive weak solutions of degenerate quasilinear elliptic systems with subcritical and critical exponents. The nonlinearities involved have semipositone and positone structures and the existence results are obtained by applying the lower and upper-solution method and variational techniques.     

An Orlicz-space approach to superlinear elliptic systems

In this paper we study superlinear elliptic systems in Hamiltonian form. Using an Orlicz-space setting, we extend the notion of critical growth to superlinear nonlinearities which do not have a polynomial growth. Existence of nontrivial solutions is proved for superlinear nonlinearities which are subcritical in this generalized sense.     

Phase plane analysis and classification of solutions of a mixed sublinear-superlinear elliptic problem

In this paper, some mixed sublinear-superlinear critical problem extending the famous problem of Brezis–Nirenberg are analysed. The existence of solutions is discussed. A phase plane analysis is performed in order to transform the problem into an ordinary differential equation. Finally, a full classification of radial solutions according to their behavior at the origin is performed for subcritical, critical and supercritical cases.     

On the Palais–Smale condition

We present a new sufficient assumption weaker than the classical Ambrosetti–Rabinowitz condition which guarantees the boundedness of (PS) sequences. Moreover, we relax the standard subcritical polynomial growth condition ensuring the compactness of a bounded (PS) sequences. We also revise the Costa–Magalhaes condition [8] to obtain Cerami condition. As a consequence, some existence results derived by minimax methods were proved. Finally, we establish the existence of positive solution under the subcritical polynomial growth condition, while the strong superlinear condition is only required along an unbounded sequence. In other words, a certain degraded oscillation is allowed.     

Concentration-compactness principle at infinity and semilinear elliptic equations involving critical and subcritical Sobolev exponents

We formulate the concentration-compactness principle at infinity for both subcritical and critical case. We show some applications to the existence theory of semilinear elliptic equations involving critical and subcritical Sobolev exponents.     

Exterior nonlinear Neumann problem

We consider the solvability of the Neumann problem for equation (1.1) in exterior domains in both cases: subcritical and critical. We establish the existence of least energy solutions. In the subcritical case the coefficient b(x) is allowed to have a potential well whose steepness is controlled by a parameter λ > 0. We show that least energy solutions exhibit a tendency to concentrate to a solution of a nonlinear problem with mixed boundary value conditions.     

SOLUTIONS OF AN ELLIPTIC EIGENVALUE PROBLEM INVOLVING SUBCRITICAL OR CRITICAL EXPONENTS

In this paper, we prove the existence of a positive solution, a negative solution and a sign-changing solution of a semilinear elliptic eigenvalue problem with constraint involving subcritical and critical Sobolev exponents. The solutions are obtained in the ω-limit sets of some descending flow curves whose starting points are specifically chosen.     

Quasilinear elliptic system with coupling on nonhomogeneous critical term

We study the existence of multiple solutions for a quasilinear elliptic system of gradient type with the possibility of coupling on the critical and subcritical terms which are not necessarily homogeneous. The solutions are obtained from a version of the Symmetric Mountain Pass Theorem. A version of the Concentration-Compactness Principle for this class of systems allows us to verify that the Palais–Smale condition is satisfied below a certain level.     

Energy critical fourth-order Schrödinger equations with subcritical perturbations

In this paper, we study the global well-posedness and scattering theory of an 8-D cubic nonlinear fourth-order Schrödinger equation, which is perturbed by a subcritical nonlinearity. We utilize the strategies in Tao et al. (2007) [16] and Zhang (2006) [17] to obtain when the cubic term is defocusing, the solution is always global no matter what the sign of the subcritical perturbation term is. Moreover, scattering will occur either when the pertubation is defocusing and 1<p<2 or when the mass of the solution is small enough and 1≤p<2.     

包含次临界和临界Sobolev指数的Schr(o)dinger-Possion方程的正解

讨论了一类包含次临界和临界Sobolev指数的Schr(o)dinger-Possion方程解的存在性.应用Nehari流形和变分方法,在不同情况下,得到了方程至少存在一个解.     

Oscillatory radial solutions for subcritical biharmonic equations

It is well known that the biharmonic equation Δ²u=u|u|^p−1 with p∈(1,∞) has positive solutions on Rn if and only if the growth of the nonlinearity is critical or supercritical. We close a gap in the existing literature by proving the existence and uniqueness, up to scaling and symmetry, of oscillatory radial solutions on Rn in the subcritical case. Analyzing the nodal properties of these solutions, we also obtain precise information about sign-changing large radial solutions and radial solutions of the Dirichlet problem on a ball.     

Existence of solution for a nonvariational elliptic system with exponential growth in dimension two

We prove existence of solution for an elliptic system on a bounded domain in dimension two. We use the Galerkin scheme in the product of Hilbert spaces. The nonlinearities may have subcritical or critical exponential growth.     

Maximum and anti-maximum principlesand eigenfunctions estimates via perturbation theory of positive solutions of elliptic equations

In this paper we discuss some new results concerning perturbation theory for second order elliptic partial differential equations related to positivity properties of such equations. We continue the study of some different notions of “small” perturbations and discuss their relations to comparisons of Green's functions, refined maximum and anti-maximum principles, ground state, and the decay of eigenfunctions. In particular, we show that if V is a positive function which is a semismall perturbation of a subcritical Schr?dinger operator H defined on a domain , and are the (Dirichlet) eigenfunctions of the equation , then for any , the function is bounded and has a continuous extension up to the Martin boundary of the pair , where is the ground state of H with a principal eigenvalue . Received: 29 November 1998     

Nontrivial solutions for resonant cooperative elliptic systems via computations of the critical groups

In this paper, we consider a class of resonant cooperative elliptic systems. Based on some new results concerning the computations of the critical groups and the Morse theory, we establish some new results about the existence and multiplicity of solutions under new classes of conditions. It turns out that our main results sharply improve some known results in the literature.     

On nontrivial solutions of critical polyharmonic elliptic systems

In the present work, we consider elliptic systems involving polyharmonic operators and critical exponents. We discuss the existence and nonexistence of nontrivial solutions to these systems. Our theorems improve and/or extend the ones established by Bartsch and Guo [T. Bartsch, Y. Guo, Existence and nonexistence results for critical growth polyharmonic elliptic systems, J. Differential Equations 220 (2006) 531-543] in both aspects of spectral interaction and regularity of lower order perturbations.     

On a fourth-order quasilinear elliptic equation of concave–convex type

We consider a fourth-order quasilinear equation depending on a nonnegative parameter λ and with subcritical or critical growth. Such equation is equivalent to a Hamiltonian system and the main goal of this work is to prove the existence of at least two positive and infinitely many solutions for such equation when the parameter λ is positive and small enough. This work was supported by FAPESP grant #07/54872-8.     

Singularly perturbed elliptic systems

We study the existence and concentration behavior of positive solutions for a class of Hamiltonian systems (two coupled nonlinear stationary Schrödinger equations). Combining the Legendre–Fenchel transformation with mountain pass theorem, we prove the existence of a family of positive solutions concentrating at a point in the limit, where related functionals realize their minimum energy. In some cases, the location of the concentration point is given explicitly in terms of the potential functions of the stationary Schrödinger equations.     

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     

Liouville-type theorems and decay estimates for solutions to higher order elliptic equations

Liouville-type theorems are powerful tools in partial differential equations. Boundedness assumptions of solutions are often imposed in deriving such Liouville-type theorems. In this paper, we establish some Liouville-type theorems without the boundedness assumption of nonnegative solutions to certain classes of elliptic equations and systems. Using a rescaling technique and doubling lemma developed recently in Polá?ik et al. (2007) [20], we improve several Liouville-type theorems in higher order elliptic equations, some semilinear equations and elliptic systems. More specifically, we remove the boundedness assumption of the solutions which is required in the proofs of the corresponding Liouville-type theorems in the recent literature. Moreover, we also investigate the singularity and decay estimates of higher order elliptic equations.     

Symmetry for extremal functions in subcritical Caffarelli–Kohn–Nirenberg inequalities

We use the formalism of the Rényi entropies to establish the symmetry range of extremal functions in a family of subcritical Caffarelli–Kohn–Nirenberg inequalities. By extremal functions we mean functions that realize the equality case in the inequalities, written with optimal constants. The method extends recent results on critical Caffarelli–Kohn–Nirenberg inequalities. Using heuristics given by a nonlinear diffusion equation, we give a variational proof of a symmetry result, by establishing a rigidity theorem: in the symmetry region, all positive critical points have radial symmetry and are therefore equal to the unique positive, radial critical point, up to scalings and multiplications. This result is sharp. The condition on the parameters is indeed complementary of the condition that determines the region in which symmetry breaking holds as a consequence of the linear instability of radial optimal functions. Compared to the critical case, the subcritical range requires new tools. The Fisher information has to be replaced by Rényi entropy powers, and since some invariances are lost, the estimates based on the Emden–Fowler transformation have to be modified.