Monotonicity of solutions of quasilinear degenerate elliptic equation in half-spaces

We prove a weak comparison principle in narrow unbounded domains for solutions to $-\Delta _p u=f(u)$ in the case $2<p< 3$ and $f(\cdot )$ is a power-type nonlinearity, or in the case $p>2$ and $f(\cdot )$ is super-linear. We exploit it to prove the monotonicity of positive solutions to $-\Delta _p u=f(u)$ in half spaces (with zero Dirichlet assumption) and therefore to prove some Liouville-type theorems.     

Asymptotic formulas for solutions of nonlocal elliptic problems

We consider nonlocal elliptic problems in plane domains and obtain asymptotic formulas for solutions in weighted spaces near junction points.     

On the index instability for some nonlocal elliptic problems

The index of unbounded operators defined on generalized solutions of nonlocal elliptic problems in plane bounded domains is investigated. It is known that nonlocal terms with smooth coefficients having zero of a certain order at the conjugation points do not affect the index of the unbounded operator. In this paper, we construct examples showing that the index may change under nonlocal perturbations with coefficients not vanishing at the points of conjugation of boundary-value conditions. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 178–193, 2007.     

Antisymmetry of solutions for some weighted elliptic problems

This article concerns the antisymmetry, uniqueness, and monotonicity properties of solutions to some elliptic functionals involving weights and a double well potential. In the one-dimensional case, we introduce the continuous odd rearrangement of an increasing function and we show that it decreases the energy functional when the weights satisfy a certain convexity-type hypothesis. This leads to the antisymmetry or oddness of increasing solutions (and not only of minimizers). We also prove a uniqueness result (which leads to antisymmetry) where a convexity-type condition by Berestycki and Nirenberg on the weights is improved to a monotonicity condition. In addition, we provide with a large class of problems where antisymmetry does not hold. Finally, some rather partial extensions in higher dimensions are also given.     

Local and global solutions for some parabolic nonlocal problems

We study local and global existence of solutions for some semilinear parabolic initial boundary value problems with autonomous nonlinearities having a “Newtonian” nonlocal term.     

Sign-changing solutions for some fourth-order nonlinear elliptic problems

In this paper, we consider the existence and multiplicity of sign-changing solutions for some fourth-order nonlinear elliptic problems and some existence and multiple are obtained. The weak solutions are sought by means of sign-changing critical theorems.     

NontriviaL solutions of some nonlinear elliptic problems

This paper is concerned with a class of semilinear elliptic Dirichlet problems approximating degenerate equations. The aim is to prove the existence of at least 4^k?1 nontrivial solutions when the degeneration set consists of k distinct connected components     

Existence of solutions for some fourth-order nonlinear elliptic problems

In this paper, we consider the existence of positive, negative and sign-changing solutions for some fourth order semilinear elliptic boundary value problems. We present new results on invariant sets of the gradient flows of the corresponding variational functionals. The structure of the invariant sets will be built into minimax procedures to construct the sign-changing solutions.     

Multiple nontrivial solutions for some fourth-order semilinear elliptic problems

In this paper, we consider the existence of multiple nontrivial solutions for some fourth order semilinear elliptic boundary value problems. The weak solutions are sought by means of Morse theory and local linking.     

On strictly positive solutions for some semilinear elliptic problems

Let B be a ball in ${\mathbb{R}^{N}}$ , N ≥ 1, let m be a possibly discontinuous and unbounded function that changes sign in B and let 0 < p < 1. We study existence and nonexistence of strictly positive solutions for semilinear elliptic problems of the form ${-\Delta u=m(x) u^{p}}$ in B, u = 0 on ?B.     

Sign-changing solutions of a class of nonlocal quasilinear elliptic boundary value problems

This paper is concerned with nonlocal Kirchhoff?s equation

     

Marcinkiewicz estimates for solutions of some elliptic problems with nonregular data

The method introduced by Ennio De Giorgi and Guido Stampacchia for the study of the regularity (L ^p , Marcinkiewicz or C ^0,α ) of the weak solutions of Dirichlet problems hinges on the handle of inequalities concerning the integral of on the subsets where |u(x)| is greater than k. In this framework, here we give a contribution with the study of the Marcinkiewicz regularity of the gradient of infinite energy solutions of Dirichlet problems with nonregular data. Dedicated to Juan Luis Vazquez for his 60th birthday (“El verano del Patriarca”, see [19]).     

Some properties of solutions of nonlocal elliptic problems in the space of generalized functions

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 11, pp. 1487–1494, November, 1989.     

Solvability of nonlocal elliptic problems in weight spaces

In this paper we consider elliptic equations of order 2m in a bounded domainQ є R ⁿ with boundaryδQ and nonlocal conditions relating the traces of the solution and its derivatives on (n − 1)-dimensional smooth manifolds Γ _i (∪ _i =∂δQ) to their values on some compact setF ⊂Q, whereF ∩δQ ≠ Φ. The Fredholm solvability of these problems in the weight spacesV _{p, a} ^/l+2m (Q) is proved for arbitrary 1<p <∞. Translated fromMatematicheskie Zametki, Vol. 67, No. 6, pp. 882–898, June, 2000. This research was supported by the Russian Foundation for Basic Research under grant No. 99-01-00028.     

Asymptotic behavior of positive solutions of some quasilinear elliptic problems

We discuss the asymptotic behavior of positive solutions ofthe quasilinear elliptic problem –_pu = a u^p–1–b(x)u^q, u| = 0, as q p – 1 + 0 and as q , via a scale argument.Here _p is the p-Laplacian with 1 < p and q > p –1. If p = 2, such problems arise in population dynamics. Ourmain results generalize the results for p = 2, but some technicaldifficulties arising from the nonlinear degenerate operator–_p are successfully overcome. As a by-product, we cansolve a free boundary problem for a nonlinear p-Laplacian equation.     

On the existence of postive solutions for some indefinite superlinear elliptic problems

In this work we show the existence and stability of positive solutions for a general calss of semilinear elliptic boundary value problems of superlinear type with indedefinite weight functions. Optimal necessary and sufficient conditions are found.     

Multipeak solutions for some singularly perturbed nonlinear elliptic problems on Riemannian manifolds

Given (M, g) a smooth compact Riemannian N-manifold, we prove that for any fixed positive integer K the problem