共查询到20条相似文献,搜索用时 31 毫秒
1.
We study uniformly elliptic fully nonlinear equations of the type F(D2u,Du,u,x)=f(x). We show that convex positively 1-homogeneous operators possess two principal eigenvalues and eigenfunctions, and study these objects; we obtain existence and uniqueness results for nonproper operators whose principal eigenvalues (in some cases, only one of them) are positive; finally, we obtain an existence result for nonproper Isaac's equations. 相似文献
2.
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. 相似文献
3.
Dario D. Monticelli 《Journal of Differential Equations》2009,247(7):1993-2026
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. 相似文献
4.
Markus Biegert Mahamadi Warma 《Nonlinear Analysis: Theory, Methods & Applications》2010,73(6):1853-698
We characterize the Lq-solvability of a class of quasi-linear elliptic equations involving the p-Laplace operator with generalized nonlinear Robin type boundary conditions on bad domains. Some uniqueness results are also given. 相似文献
5.
D. Castorina 《Nonlinear Analysis: Theory, Methods & Applications》2010,72(5):2195-2203
Extending a previous result of Tang [1] we prove the uniqueness of positive radial solutions of Δpu+f(u)=0, subject to Dirichlet boundary conditions on an annulus in Rn with 2<p≤n, under suitable hypotheses on the nonlinearity f. This argument also provides an alternative proof for the uniqueness of positive solutions of the same problem in a finite ball (see [9]), in the complement of a ball or in the whole space Rn (see [10], [3] and [11]). 相似文献
6.
Let Ω be a bounded domain in R2, u+=u if u?0, u+=0 if u<0, u−=u+−u. In this paper we study the existence of solutions to the following problem arising in the study of a simple model of a confined plasma
7.
Yihong Du 《Journal of Differential Equations》2008,244(1):117-169
Let u? be a single layered radially symmetric unstable solution of the Allen-Cahn equation −?2Δu=u(u−a(|x|))(1−u) over the unit ball with Neumann boundary conditions. Based on our estimate of the small eigenvalues of the linearized eigenvalue problem at u? when ? is small, we construct solutions of the form u?+v?, with v? non-radially symmetric and close to zero in the unit ball except near one point x0 such that |x0| is close to a nondegenerate critical point of a(r). Such a solution has a sharp layer as well as a spike. 相似文献
8.
Weilin Zou 《Nonlinear Analysis: Theory, Methods & Applications》2010,73(9):3069-3082
This paper deals with a class of degenerate quasilinear elliptic equations of the form −div(a(x,u,∇u)=g−div(f), where a(x,u,∇u) is allowed to be degenerate with the unknown u. We prove existence of bounded solutions under some hypothesis on f and g. Moreover we prove that there exists a renormalized solution in the case where g∈L1(Ω) and f∈(Lp′(Ω))N. 相似文献
9.
Nicu?or Costea 《Nonlinear Analysis: Theory, Methods & Applications》2009,71(9):4271-4278
We study the problem in Ω, u=0 on ∂Ω, where Ω is a bounded domain in RN, is a continuous function and λ and ε are two positive constants. We prove that for any ε>0 each λ∈(0,λ1) is an eigenvalue of the above problem, where λ1 is the principal eigenvalue of the Laplace operator on Ω. Moreover, for each eigenvalue λ∈(0,λ1) it corresponds a unique eigenfunction. The proofs will be based on the Banach fixed point theorem combined with adequate variational techniques. 相似文献
10.
In this paper we consider radially symmetric solutions of the nonlinear Dirichlet problem Δu+f(|x|,u)=0 in Ω, where Ω is a ball in RN, N?3 and f satisfies some appropriate assumptions. We prove existence of radially symmetric solutions with k prescribed number of zeros. Moreover, when f(|x|,u)=K(|x|)|u|p−1u, using the uniqueness result due to Tanaka (2008) [21], we verify that these solutions are non-degenerate and we prove that their radial Morse index is exactly k. 相似文献
11.
Biagio Ricceri 《Nonlinear Analysis: Theory, Methods & Applications》2009,71(9):4151-4157
If X is a real Banach space, we denote by WX the class of all functionals possessing the following property: if {un} is a sequence in X converging weakly to u∈X and lim infn→∞Φ(un)≤Φ(u), then {un} has a subsequence converging strongly to u.In this paper, we prove the following result:Let X be a separable and reflexive real Banach space; an interval; a sequentially weakly lower semicontinuous C1 functional, belonging to WX, bounded on each bounded subset of X and whose derivative admits a continuous inverse on X∗; a C1 functional with compact derivative. Assume that, for each λ∈I, the functional Φ−λJ is coercive and has a strict local, not global minimum, say .Then, for each compact interval [a,b]⊆I for which , there exists r>0 with the following property: for every λ∈[a,b] and every C1 functional with compact derivative, there exists δ>0 such that, for each μ∈[0,δ], the equation
Φ′(x)=λJ′(x)+μΨ′(x) 相似文献
12.
Ognjen Milatovic 《Differential Geometry and its Applications》2004,21(3):361-377
We consider a family of Schrödinger-type differential expressions L(κ)=D2+V+κV(1), where κ∈C, and D is the Dirac operator associated with a Clifford bundle (E,∇E) of bounded geometry over a manifold of bounded geometry (M,g) with metric g, and V and V(1) are self-adjoint locally integrable sections of EndE. We also consider the family I(κ)=*(∇F)∇F+V+κV(1), where κ∈C, and ∇F is a Hermitian connection on a Hermitian vector bundle F of bonded geometry over a manifold of bounded geometry (M,g), and V and V(1) are self-adjoint locally integrable sections of EndF. We give sufficient conditions for L(κ) and I(κ) to have a realization in L2(E) and L2(F), respectively, as self-adjoint holomorphic families of type (B). In the proofs we use Kato's inequality for Bochner Laplacian operator and Weitzenböck formula. 相似文献
13.
We prove an optimal dispersive L∞ decay estimate for a three-dimensional wave equation perturbed with a large nonsmooth potential belonging to a particular Kato class. The proof is based on a spectral representation of the solution and suitable resolvent estimates for the perturbed operator. 相似文献
14.
In this paper, we establish a Gromoll-Meyer splitting theorem and a shifting theorem for J∈C2-0(E,R) and by using the finite-dimensional approximation, mollifiers and Morse theory we generalize the Poincaré-Hopf theorem to J∈C1(E,R) case. By combining the Poincaré-Hopf theorem and the splitting theorem, we study the existence of multiple solutions for jumping nonlinear elliptic equations. 相似文献
15.
We consider autonomous parabolic Dirichlet problems in a regular unbounded open set Ω⊂RN involving second-order operator A with (possibly) unbounded coefficients. We determine new conditions on the coefficients of A yielding global gradient estimates for the bounded classical solution. 相似文献
16.
17.
Dumitru Motreanu 《Journal of Differential Equations》2007,232(1):1-35
In this paper we examine semilinear and nonlinear Neumann problems with a nonsmooth locally Lipschitz potential function. Using variational methods based on the nonsmooth critical point theory, for the semilinear problem we prove a multiplicity result under conditions of double resonance at higher eigenvalues. Our proof involves a nonsmooth extension of the reduction method due to Castro-Lazer-Thews. The nonlinear problem is driven by the p-Laplacian. So first we make some observations about the beginning of the spectrum of (−Δp,W1,p(Z)). Then we prove an existence and multiplicity result. The existence result permits complete double resonance. The multiplicity result specialized in the semilinear case (i.e. p=2) corresponds to the super-sub quadratic situation. 相似文献
18.
Su Jiabao 《数学学报(英文版)》1998,14(3):411-418
In this paper we study the existence of nontrivial solutions of a class of asymptotically linear elliptic resonant problems
at higher eigenvalues with the nonlinear term which may be unbounded by making use of the Morse theory for aC
2-function at both isolated critical point and infinity. 相似文献
19.
We extend to the case of many competing densities the results of the paper (Ann. Inst. H. Poincaré 6 (2002)). More precisely, we are concerned with an optimal partition problem in N-dimensional domains related to the method of nonlinear eigenvalues introduced by Nehari, (Acta Math. 105 (1961)). We prove existence of the minimal partition and some extremality conditions. Moreover, in the case of two-dimensional domains we give an asymptotic formula near the multiple intersection points. Finally, we show some connections between the variational problem and the behavior of competing species systems with large interaction. 相似文献
20.
The purpose of this work is to solve exterior problems in the half-space for the Laplace operator. We give existence and unicity results in weighted Lp's theory with 1<p<∞. This paper extends the studies done in [C. Amrouche, V. Girault, J. Giroire, Dirichlet and Neumann exterior problems for the n-dimensional Laplace operator, an approach in weighted Sobolev spaces, J. Math. Pures Appl. 76 (1) (1997) 55-81] with Dirichlet and Neumann conditions. 相似文献