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1.
We consider here a class of nonlinear Dirichlet problems, in a bounded domain , of the form
investigating the problem of uniqueness of solutions. The functions (s) and
satisfy rather general assumptions of locally Lipschitz continuity (with possibly exponential growth) and the datum f is in L1(). Uniqueness of solutions is proved both for coercive a(x, s) and for the case of a(x, s) degenerating for s large. 相似文献
2.
Dimitri Mugnai 《Calculus of Variations and Partial Differential Equations》2008,32(4):481-497
We show that a semilinear Dirichlet problem in bounded domains of in presence of subcritical exponential nonlinearities has four nontrivial solutions near resonance.
Research supported by the Italian National Project Metodi Variazionali ed Equazioni Differenziali Non Lineari. 相似文献
3.
Andrey Shishkov Laurent Véron 《Calculus of Variations and Partial Differential Equations》2008,33(3):343-375
We study the limit behaviour of solutions of with initial data k
δ
0 when k → ∞, where h is a positive nondecreasing function and p > 1. If h(r) = r
β
, β > N(p − 1) − 2, we prove that the limit function u
∞ is an explicit very singular solution, while such a solution does not exist if β ≤ N(p − 1) − 2. If lim
inf
r→ 0
r
2 ln (1/h(r)) > 0, u
∞ has a persistent singularity at (0, t) (t ≥ 0). If , u
∞ has a pointwise singularity localized at (0, 0). 相似文献
4.
Luis Caffarelli Yan Yan Li Louis Nirenberg 《Journal of Fixed Point Theory and Applications》2009,5(2):353-395
We study strong maximum principles for singular solutions of nonlinear elliptic and degenerate elliptic equations of second
order. An application is given on symmetry of positive solutions in a punctured ball using the method of moving planes.
Dedicated to Felix Browder on his 80th birthday 相似文献
5.
Given (M, g) a smooth compact Riemannian N-manifold, we prove that for any fixed positive integer K the problem
has a K-peaks solution, whose peaks collapse, as ε goes to zero, to an isolated local minimum point of the scalar curvature. Here p > 2 if N = 2 and .
E. N. Dancer was partially supported by the ARC. A. M. Micheletti and A. Pistoia are supported by Mi.U.R. Project “Metodi
variazionali e topologici nello studio di fenomeni non lineari”. 相似文献
6.
We use the basic formulation of Ekeland’s variational principle to establish characterizations of complete path metric spaces
which, being described in terms of the strong slope, are called coherent as in [3]. We also provide some basic nonlinear error bound and metric regularity results, in the context of coherent spaces. 相似文献
7.
N. M. Ivochkina 《Journal of Fixed Point Theory and Applications》2008,4(1):47-56
We adapt to degenerate m-Hessian evolution equations the notion of m-approximate solutions introduced by N. Trudinger for m-Hessian elliptic equations, and we present close to necessary and sufficient conditions guaranteeing the existence and uniqueness
of such solutions for the first initial boundary value problem.
Dedicated to Professor Felix Browder 相似文献
8.
We prove the existence of an unbounded sequence of solutions for an elliptic equation of the form \({-\Delta u=\lambda u + a(x)g(u)+f(x), u\in H^1_0(\Omega)}\), where \({\lambda \in \mathbb{R}, g(\cdot)}\) is subcritical and superlinear at infinity, and a(x) changes sign in Ω; moreover, g( ? s) = ? g(s) \({\forall s}\). The proof uses Rabinowitz’s perturbation method applied to a suitably truncated problem; subsequent energy and Morse index estimates allow us to recover the original problem. We consider the case of \({\Omega\subset \mathbb{R}^N}\) bounded as well as \({\Omega=\mathbb{R}^N, \, N\geqslant 3}\). 相似文献
9.
Francesca Alessio Piero Montecchiari 《Calculus of Variations and Partial Differential Equations》2007,30(1):51-83
We consider a class of semilinear elliptic equations of the form
where is a periodic, positive function and is modeled on the classical two well Ginzburg-Landau potential . We show, via variational methods, that if the set of solutions to the one dimensional heteroclinic problem
has a discrete structure, then (0.1) has infinitely many solutions periodic in the variable y and verifying the asymptotic conditions as uniformly with respect to .
Supported by MURST Project ‘Metodi Variazionali ed Equazioni Differenziali Non Lineari’. 相似文献
10.
Arrigo Cellina Mihai Vornicescu 《Calculus of Variations and Partial Differential Equations》2009,35(2):263-270
In this paper we establish an existence and regularity result for solutions to the problem
for boundary data that are constant on each connected component of the boundary of Ω. The Lagrangean L belongs to a class that contains both extended valued Lagrangeans and Lagrangeans with linear growth. Regularity means that
the solution is Lipschitz continuous and that, in addition, is bounded. 相似文献
11.
P. Quittner W. Reichel 《Calculus of Variations and Partial Differential Equations》2008,32(4):429-452
Consider the equation −Δu = 0 in a bounded smooth domain , complemented by the nonlinear Neumann boundary condition ∂ν
u = f(x, u) − u on ∂Ω. We show that any very weak solution of this problem belongs to L
∞(Ω) provided f satisfies the growth condition |f(x, s)| ≤ C(1 + |s|
p
) for some p ∈ (1, p*), where . If, in addition, f(x, s) ≥ −C + λs for some λ > 1, then all positive very weak solutions are uniformly a priori bounded. We also show by means of examples that
p* is a sharp critical exponent. In particular, using variational methods we prove the following multiplicity result: if N ∈ {3, 4} and f(x, s) = s
p
then there exists a domain Ω and such that our problem possesses at least two positive, unbounded, very weak solutions blowing up at a prescribed point of
∂Ω provided . Our regularity results and a priori bounds for positive very weak solutions remain true if the right-hand side in the differential
equation is of the form h(x, u) with h satisfying suitable growth conditions. 相似文献
12.
Zhaoli Liu Jiabao Su Zhi-Qiang Wang 《Calculus of Variations and Partial Differential Equations》2009,35(4):463-480
In this paper, we study existence of nontrivial solutions to the elliptic equation
and to the elliptic system
where Ω is a bounded domain in with smooth boundary ∂Ω, , f (x, 0) = 0, with m ≥ 2 and . Nontrivial solutions are obtained in the case in which the nonlinearities have linear growth. That is, for some c > 0, for and , and for and , where I
m
is the m × m identity matrix. In sharp contrast to the existing results in the literature, we do not make any assumptions at infinity
on the asymptotic behaviors of the nonlinearity f and .
Z. Liu was supported by NSFC(10825106, 10831005). J. Su was supported by NSFC(10831005), NSFB(1082004), BJJW-Project(KZ200810028013)
and the Doctoral Programme Foundation of NEM of China (20070028004). 相似文献
13.
Giovanna Cerami Mónica Clapp 《Calculus of Variations and Partial Differential Equations》2007,30(3):353-367
We prove the existence of a sign changing solution to the semilinear elliptic problem , in an exterior domain Ω having finite symmetries. 相似文献
14.
Ram U. Verma 《Positivity》2009,13(4):771-782
First, based on η-maximal accretiveness, a generalization to Rockafellar’s theorem (1976) in the context of approximating a solution to a general
inclusion problem involving a multivalued η-maximal accretive mapping using the proximal point algorithm in a q-uniformly smooth Banach space setting is considered.
Then an application to a minimization problem of a functional is examined. The general framework for η-maximal accretiveness generalizes the general theory of multivalued maximal monotone mappings.
相似文献
15.
Veronica Felli Emmanuel Hebey Frédéric Robert 《NoDEA : Nonlinear Differential Equations and Applications》2005,12(2):171-213
Given (M,g) a smooth compact Riemannian manifold of dimension n ≥ 5, we consider equations like
where
is a Paneitz-Branson type operator with constant coefficients α and aα, u is required to be positive, and
is critical from the Sobolev viewpoint. We define the energy function Em as the infimum of
over the u’s which are solutions of the above equation. We prove that Em (α ) →+∞ as α →+∞ . In particular, for any Λ > 0, there exists α0 > 0 such that for α ≥ α0, the above equation does not have a solution of energy less than or equal to Λ. 相似文献
16.
Changjiang Zhu Zhian Wang 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2004,55(6):994-1014
In this paper, we study the global existence and the asymptotic behavior of the solutions to the Cauchy problem for the following nonlinear evolution equations with ellipticity and dissipative effects
with initial data
where and are positive constants such that < 1, < (1–). Through constructing a correct function
defined by (2.13) and using the energy method, we show
as
and the solutions decay with exponential rates. The same problem is studied by Tang and Zhao [10] for the case of (±, ±) = (0,0).Received: November 18, 2003 相似文献
((E)) |
((I)) |
17.
Stefan Krömer 《Calculus of Variations and Partial Differential Equations》2008,32(2):219-236
We study functionals of the form
where u is a real valued function over the ball which vanishes on the boundary and W is nonconvex. The functional is assumed to be radially symmetric in the sense that W only depends on . Existence of one and radial symmetry of all global minimizers is shown with an approach based on convex relaxation. Our
assumptions on G do not include convexity, thus extending a result of A. Cellina and S. Perrotta. 相似文献
18.
Zhian Wang 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2006,57(3):399-418
We derive the optimal decay rates of solution to the Cauchy problem for a set of nonlinear evolution equations with ellipticity
and dissipative effects
with initial data
where α and ν are positive constants such that α < 1, ν < α(1 − α), which is a special case of (1.1). We show that the solution
to the system decays with the same rate to that of its associated homogenous linearized system. The main results are obtained
by the use of Fourier analysis and interpolation inequality under some suitable restrictions on coefficients α and ν. Moreover,
we discuss the asymptotic behavior of the solution to general system (1.1) at the end.
The research was supported by the F. S. Chia Scholarship of the University of Alberta.
Received: January 27, 2005; revised: April 27, 2005 相似文献
19.
Futoshi Takahashi 《Archiv der Mathematik》2009,93(2):191-197
In this note, we consider the problem
on a smooth bounded domain Ω in for p > 1. Let u
p
be a positive solution of the above problem with Morse index less than or equal to . We prove that if u
p
further satisfies the assumption as p → ∞, then the number of maximum points of u
p
is less than or equal to m for p sufficiently large. If Ω is convex, we also show that a solution of Morse index one satisfying the above assumption has a
unique critical point and the level sets are star-shaped for p sufficiently large.
相似文献
20.
We consider the mixed problem,
in a class of Lipschitz graph domains in two dimensions with Lipschitz constant at most 1. We suppose the Dirichlet data,
f
D
, has one derivative in L
p
(D) of the boundary and the Neumann data, f
N
, is in L
p
(N). We find a p
0 > 1 so that for p in an interval (1, p
0), we may find a unique solution to the mixed problem and the gradient of the solution lies in L
p
.
L. Lanzani, L. Capogna and R. M. Brown were supported, in part, by the U.S. National Science Foundation. 相似文献