R2 中带不定权与临界位势的非线性椭圆问题

该文讨论了R²中一类带不定权且含临界位势的二阶椭圆方程的特征值问题,然后,借此特征值问题的第一特征值性质,利用临界点理论及Trudinger-Moser不等式,证明了R² 中一类带不定权且含临界位势的二阶非线性椭圆方程非平凡解的存在性.     

The effects of indefinite nonlinear boundary conditions on the structure of the positive solutions set of a logistic equation

We investigate a semilinear elliptic equation with a logistic nonlinearity and an indefinite nonlinear boundary condition, both depending on a parameter λ. Overall, we analyze the effect of the indefinite nonlinear boundary condition on the structure of the positive solutions set. Based on variational and bifurcation techniques, our main results establish the existence of three nontrivial non-negative solutions for some values of λ, as well as their asymptotic behavior. These results suggest that the positive solutions set contains an S-shaped component in some case, as well as a combination of a C-shaped and an S-shaped components in another case.     

Existence results for semilinear elliptic variational inequalities with changing sign nonlinearities

In this paper we present some existence results for a class of semilinear elliptic variational inequalities, depending on a real parameter λ, with changing sign nonlinearities. The fundamental tool to prove the existence result is a penalization method combined with the Mountain Pass Theorem and the Linking Theorem, respectively in the case λ < λ ₁ and λ ≥ λ ₁, where λ₁ is the first eigenvalue of the uniformly elliptic operator A involved in the variational inequality.     

Structure of boundary blow-up solutions for quasi-linear elliptic problems II: small and intermediate solutions

The structure of positive boundary blow-up solutions to quasi-linear elliptic problems of the form −Δ_pu=λf(u), u=∞ on ∂Ω, 1<p<∞, is studied in a bounded smooth domain , for a class of nonlinearities f∈C¹((0,∞)?{z₂})∩C⁰[0,∞) satisfying f(0)=f(z₁)=f(z₂)=0 with 0<z₁<z₂, f<0 in (0,z₁)∪(z₂,∞), f>0 in (z₁,z₂). Large, small and intermediate solutions are obtained for λ sufficiently large. It is known from Part I (see Structure of boundary blow-up solutions for quasilinear elliptic problems, part (I): large and small solutions, preprint), that the large solution is the unique large solution to the problem. We will see that the small solution is also the unique small solution to the problem while there are infinitely many intermediate solutions. Our results are new even for the case p=2.     

Multiplicity of solutions for a quasilinear elliptic problem via the cohomological index

In this paper we study a quasilinear elliptic problem with the Dirichlet boundary condition in a bounded domain involving the operator Au=−Δ_pu−Δ_qu. Assuming that the nonlinearity has a concave-convex behavior we obtain some multiplicity results. More precisely, we obtain five nontrivial solutions; two by a minimization argument, two by the mountain pass theorem, and the other by a cohomological linking theorem.     

Constant-sign and sign-changing solutions for nonlinear eigenvalue problems

We prove the existence of multiple constant-sign and sign-changing solutions for a nonlinear elliptic eigenvalue problem under Dirichlet boundary condition involving the p$p$-Laplacian. More precisely, we establish the existence of a positive solution, of a negative solution, and of a nontrivial sign-changing solution when the eigenvalue parameter λ$\lambda$ is greater than the second eigenvalue _λ2${\lambda }_2$ of the negative p$p$-Laplacian, extending results by Ambrosetti–Lupo, Ambrosetti–Mancini, and Struwe. Our approach relies on a combined use of variational and topological tools (such as, e.g., critical points, Mountain-Pass theorem, second deformation lemma, variational characterization of the first and second eigenvalue of the p$p$-Laplacian) and comparison arguments for nonlinear differential inequalities. In particular, the existence of extremal nontrivial constant-sign solutions plays an important role in the proof of sign-changing solutions.     

Positive solutions of quasilinear parabolic systems with Dirichlet boundary condition

Coupled systems for a class of quasilinear parabolic equations and the corresponding elliptic systems, including systems of parabolic and ordinary differential equations are investigated. The aim of this paper is to show the existence, uniqueness, and asymptotic behavior of time-dependent solutions. Also investigated is the existence of positive maximal and minimal solutions of the corresponding quasilinear elliptic system. The elliptic operators in both systems are allowed to be degenerate in the sense that the density-dependent diffusion coefficients Di(ui) may have the property Di(0)=0 for some or all i=1,…,N, and the boundary condition is ui=0. Using the method of upper and lower solutions, we show that a unique global classical time-dependent solution exists and converges to the maximal solution for one class of initial functions and it converges to the minimal solution for another class of initial functions; and if the maximal and minimal solutions coincide then the steady-state solution is unique and the time-dependent solution converges to the unique solution. Applications of these results are given to three model problems, including a scalar polynomial growth problem, a coupled system of polynomial growth problem, and a two component competition model in ecology.     

On elliptic problems involving critical Hardy-Sobolev exponents and sign-changing function

In this paper, we deal with the existence and nonexistence of nonnegative nontrivial weak solutions for a class of degenerate quasilinear elliptic problems with weights and nonlinearity involving the critical Hardy-Sobolev exponent and a sign-changing function. Some existence results are obtained by splitting the Nehari manifold and by exploring some properties of the best Hardy-Sobolev constant together with an approach developed by Brezis and Nirenberg.     

An elliptic system with bifurcation parameters on the boundary conditions

In this paper we consider the elliptic system Δu=a(x)upvq, Δv=b(x)urvs in Ω, a smooth bounded domain, with boundary conditions , on ∂Ω. Here λ and μ are regarded as parameters and p,s>1, q,r>0 verify (p−1)(s−1)>qr. We consider the case where a(x)?0 in Ω and a(x) is allowed to vanish in an interior subdomain Ω₀, while b(x)>0 in . Our main results include existence of nonnegative nontrivial solutions in the range 0<λ<λ₁?∞, μ>0, where λ₁ is characterized by means of an eigenvalue problem, and the uniqueness of such solutions. We also study their asymptotic behavior in all possible cases: as both λ,μ→0, as λ→λ₁<∞ for fixed μ (respectively μ→∞ for fixed λ) and when both λ,μ→∞ in case λ₁=∞.     

Gradient regularity for elliptic equations in the Heisenberg group

We give dimension-free regularity conditions for a class of possibly degenerate sub-elliptic equations in the Heisenberg group exhibiting super-quadratic growth in the horizontal gradient; this solves an issue raised in [J.J. Manfredi, G. Mingione, Regularity results for quasilinear elliptic equations in the Heisenberg group, Math. Ann. 339 (2007) 485-544], where only dimension dependent bounds for the growth exponent are given. We also obtain explicit a priori local regularity estimates, and cover the case of the horizontal p-Laplacean operator, extending some regularity proven in [A. Domokos, J.J. Manfredi, C1,α-regularity for p-harmonic functions in the Heisenberg group for p near 2, in: Contemp. Math., vol. 370, 2005, pp. 17-23]. In turn, using some recent techniques of Caffarelli and Peral [L. Caffarelli, I. Peral, On W1,p estimates for elliptic equations in divergence form, Comm. Pure Appl. Math. 51 (1998) 1-21], the a priori estimates found are shown to imply the suitable local Calderón-Zygmund theory for the related class of non-homogeneous, possibly degenerate equations involving discontinuous coefficients. These last results extend to the sub-elliptic setting a few classical non-linear Euclidean results [T. Iwaniec, Projections onto gradient fields and Lp-estimates for degenerated elliptic operators, Studia Math. 75 (1983) 293-312; E. DiBenedetto, J.J. Manfredi, On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems, Amer. J. Math. 115 (1993) 1107-1134], and to the non-linear case estimates of the same nature that were available in the sub-elliptic setting only for solutions to linear equations.     

Soliton solutions for quasilinear Schrödinger equations with critical growth

In this paper we establish the existence of standing wave solutions for quasilinear Schrödinger equations involving critical growth. By using a change of variables, the quasilinear equations are reduced to semilinear one, whose associated functionals are well defined in the usual Sobolev space and satisfy the geometric conditions of the mountain pass theorem. Using this fact, we obtain a Cerami sequence converging weakly to a solution v. In the proof that v is nontrivial, the main tool is the concentration-compactness principle due to P.L. Lions together with some classical arguments used by H. Brezis and L. Nirenberg (1983) in [9].     

Singular quasilinear elliptic problems with indefinite weights and critical potential

The Dirichlet problem for a quasilinear sub-critical inhomogeneous elliptic equation with critical potential and singular coefficients, which has indefinite weights in RN , is studied in this paper. We discuss the corresponding eigenvalue problems by the variational techniques and Picone’s identity, and obtain the existence of non-trivial solutions for the inhomogeneous Dirichlet problem by using Hardy inequality, Mountain Pass Lemma in conjunction with the property of eigenvalues.     

A nondifferentiable extension of a theorem of Pucci and Serrin and applications

We study the multiplicity of critical points for functionals which are only differentiable along some directions. We extend to this class of functionals the three critical point theorem of Pucci and Serrin and we apply it to a one-parameter family of functionals Jλ, λ∈I⊂R. Under suitable assumptions, we locate an open subinterval of values λ in I for which Jλ possesses at least three critical points. Applications to quasilinear boundary value problems are also given.     

A uniqueness result for a semilinear elliptic problem: A computer-assisted proof

Starting with the famous article [A. Gidas, W.M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979) 209-243], many papers have been devoted to the uniqueness question for positive solutions of −Δu=λu+up in Ω, u=0 on ∂Ω, where p>1 and λ ranges between 0 and the first Dirichlet eigenvalue λ₁(Ω) of −Δ. For the case when Ω is a ball, uniqueness could be proved, mainly by ODE techniques. But very little is known when Ω is not a ball, and then only for λ=0. In this article, we prove uniqueness, for all λ∈[0,λ₁(Ω)), in the case Ω=²(0,1) and p=2. This constitutes the first positive answer to the uniqueness question in a domain different from a ball. Our proof makes heavy use of computer assistance: we compute a branch of approximate solutions and prove existence of a true solution branch close to it, using fixed point techniques. By eigenvalue enclosure methods, and an additional analytical argument for λ close to λ₁(Ω), we deduce the non-degeneracy of all solutions along this branch, whence uniqueness follows from the known bifurcation structure of the problem.     

Indefinite quasilinear elliptic problems with subcritical and supercritical nonlinearities on unbounded domains

By using the fibering method, we study the existence of non-negative solutions for a class of indefinite quasilinear elliptic problems on unbounded domains with noncompact boundary, in the presence of competing subcritical and supercritical lower order nonlinearities.     

Maximum principles for weak solutions of degenerate elliptic equations with a uniformly elliptic direction

For second order linear equations and inequalities which are degenerate elliptic but which possess a uniformly elliptic direction, we formulate and prove weak maximum principles which are compatible with a solvability theory in suitably weighted versions of L²-based Sobolev spaces. The operators are not necessarily in divergence form, have terms of lower order, and have low regularity assumptions on the coefficients. The needed weighted Sobolev spaces are, in general, anisotropic spaces defined by a non-negative continuous matrix weight. As preparation, we prove a Poincaré inequality with respect to such matrix weights and analyze the elementary properties of the weighted spaces. Comparisons to known results and examples of operators which are elliptic away from a hyperplane of arbitrary codimension are given. Finally, in the important special case of operators whose principal part is of Grushin type, we apply these results to obtain some spectral theory results such as the existence of a principal eigenvalue.     

An asymmetric Neumann problem with weights

We prove the existence of a first nonprincipal eigenvalue for an asymmetric Neumann problem with weights involving the p-Laplacian (cf. (1.2) below). As an application we obtain a first nontrivial curve in the corresponding Fu?ik spectrum (cf. (1.4) below). The case where one of the weights has meanvalue zero requires some special attention in connexion with the (PS) condition and with the mountain pass geometry.     

含距离位势的拟线性椭圆方程解的存在性

变分原理证明了一类含距离位势的拟线性椭圆方程齐次Dirichlet边界条件下第一特征值问题的可解性.进一步,利用临界点理论得到了一类含距离位势的非线性椭圆方程非平凡解的存在性.     

Bounded positive entire solutions of singular quasilinear elliptic equations

Suppose that β?0 is a constant and that is a continuous function with R₊:=(0,∞). This paper investigates N-dimensional singular, quasilinear elliptic equations of the form