共查询到20条相似文献,搜索用时 46 毫秒
1.
Marius Ghergu 《Journal of Differential Equations》2003,195(2):520-536
We establish several existence and nonexistence results for the boundary value problem −Δu+K(x)g(u)=λf(x,u)+μh(x) in Ω, u=0 on ∂Ω, where Ω is a smooth bounded domain in , λ and μ are positive parameters, h is a positive function, while f has a sublinear growth. The main feature of this paper is that the nonlinearity g is assumed to be unbounded around the origin. Our analysis shows the importance of the role played by the decay rate of g combined with the signs of the extremal values of the potential K(x) on . The proofs are based on various techniques related to the maximum principle for elliptic equations. 相似文献
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Hongtao Xue 《Journal of Mathematical Analysis and Applications》2011,384(2):439-443
By a sub-supersolution method and a perturbed argument, we improve the earlier results concerning the existence of ground state solutions to a semilinear elliptic problem −Δu+p(x)q|∇u|=f(x,u), u>0, x∈RN, , where q∈(1,2], for some α∈(0,1), p(x)?0, ∀x∈RN, and f:RN×(0,∞)→[0,∞) is a locally Hölder continuous function which may be singular at zero. 相似文献
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Marcelo Montenegro Olivâine Santana de Queiroz 《Journal of Differential Equations》2009,246(2):482-511
We study the nonlinear elliptic problem −Δu=χ{u>0}(logu+λf(x,u)) in Ω⊂Rn with u=0 on ∂Ω. The function is nondecreasing, sublinear and fu is continuous. For every λ>0, we obtain a maximal solution uλ?0 and prove its global regularity . There is a constant λ∗ such that uλ vanishes on a set of positive measure for 0<λ<λ∗, and uλ>0 for λ>λ∗. If f is concave, for λ>λ∗ we characterize uλ by its stability. 相似文献
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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
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Kyungkeun Kang 《Journal of Differential Equations》2011,251(9):2466-2493
We study the 3×3 elliptic systems ∇(a(x)∇×u)−∇(b(x)∇⋅u)=f, where the coefficients a(x) and b(x) are positive scalar functions that are measurable and bounded away from zero and infinity. We prove that weak solutions of the above system are Hölder continuous under some minimal conditions on the inhomogeneous term f. We also present some applications and discuss several related topics including estimates of the Green?s functions and the heat kernels of the above systems. 相似文献
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We study the stability of conservative solutions of the Cauchy problem for the Camassa-Holm equation ut−uxxt+κux+3uux−2uxuxx−uuxxx=0 with periodic initial data u0. In particular, we derive a new Lipschitz metric dD with the property that for two solutions u and v of the equation we have dD(u(t),v(t))?eCtdD(u0,v0). The relationship between this metric and usual norms in and is clarified. 相似文献
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Hernán R. Henríquez 《Nonlinear Analysis: Theory, Methods & Applications》2009,71(12):6029-6037
Given a∈L1(R) and the generator A of an L1-integrable resolvent family of linear bounded operators defined on a Banach space X, we prove the existence of compact almost automorphic solutions of the semilinear integral equation for each f:R×X→X compact almost automorphic in t, for each x∈X, and satisfying Lipschitz and Hölder type conditions. In the scalar linear case, we prove that a∈L1(R) positive, nonincreasing and log-convex is sufficient to obtain the existence of compact almost automorphic solutions. 相似文献
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Zongming Guo 《Journal of Differential Equations》2005,211(1):187-217
The structure of positive boundary blow-up solutions to quasi-linear elliptic problems of the form −Δpu=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. 相似文献
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We consider the first initial boundary value problem for the non-autonomous nonclassical diffusion equation ut−εΔut−Δu+f(u)=g(t), ε∈[0,1], in a bounded domain in RN. Under a Sobolev growth rate of the nonlinearity f and a suitable exponential growth of the external force g, using the asymptotic a priori estimate method, we prove the existence of pullback D-attractors in the space and the upper semicontinuity of at ε=0. 相似文献
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Wolfgang Reichel 《Journal of Differential Equations》2010,248(7):1866-680
We consider the 2m-th order elliptic boundary value problem Lu=f(x,u) on a bounded smooth domain Ω⊂RN with Dirichlet boundary conditions on ∂Ω. The operator L is a uniformly elliptic linear operator of order 2m whose principle part is of the form . We assume that f is superlinear at the origin and satisfies , , where are positive functions and q>1 is subcritical. By combining degree theory with new and recently established a priori estimates, we prove the existence of a nontrivial solution. 相似文献
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We study the Cauchy problem for a class of p-evolution operators P(t,x,Dt,Dx) in , with less than coefficients with respect to the time variable.According to Lipschitz, log-lipschitz or Hölder regularity we find well-posedness in Sobolev spaces or in Gevrey classes. 相似文献
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The existence and concentration behavior of nodal solutions are established for the equation −?2Δu+V(z)u=f(u) in Ω, where Ω is a domain in R2, not necessarily bounded, V is a positive Hölder continuous function and f∈C1 is an odd function having critical exponential growth. 相似文献
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Alfredo Cano 《Journal of Differential Equations》2007,237(1):133-158
We consider the problem −Δu+a(x)u=f(x)|u|2*−2u in Ω, u=0 on ∂Ω, where Ω is a bounded smooth domain in RN, N?4, is the critical Sobolev exponent, and a,f are continuous functions. We assume that Ω, a and f are invariant under the action of a group of orthogonal transformations. We obtain multiplicity results which contain information about the symmetry and symmetry-breaking properties of the solutions, and about their nodal domains. Our results include new multiplicity results for the Brezis-Nirenberg problem −Δu+λu=|u|2*−2u in Ω, u=0 on ∂Ω. 相似文献
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Julián López-Gómez 《Journal of Differential Equations》2006,224(2):385-439
In this paper, we prove some optimal uniqueness results for large solutions of a canonical class of semilinear equations under minimal regularity conditions on the weight function in front of the non-linearity and combine these results with the localization method introduced in [López-Gómez, The boundary blow-up rate of large solutions, J. Differential Equations 195 (2003) 25-45] to prove that any large solution L of Δu=a(x)up, p>1, a>0, must satisfy
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Jie Xiao 《Journal of Differential Equations》2006,224(2):277-295
Let u(t,x) be the solution of the heat equation (∂t-Δx)u(t,x)=0 on subject to u(0,x)=f(x) on Rn. The main goal of this paper is to characterize such a nonnegative measure μ on that f(x)?u(t2,x) induces a bounded embedding from the Sobolev space , p∈[1,n) into the Lebesgue space , q∈(0,∞). 相似文献
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We consider the Dirichlet problem for positive solutions of the equation −Δm(u)=f(u) in a bounded smooth domain Ω, with f locally Lipschitz continuous, and prove some regularity results for weak solutions. In particular when f(s)>0 for s>0 we prove summability properties of , and Sobolev's and Poincaré type inequalities in weighted Sobolev spaces with weight |Du|m−2. The point of view of considering |Du|m−2 as a weight is particularly useful when studying qualitative properties of a fixed solution. In particular, exploiting these new regularity results we can prove a weak comparison principle for the solutions and, using the well known Alexandrov-Serrin moving plane method, we then prove a general monotonicity (and symmetry) theorem for positive solutions u of the Dirichlet problem in bounded (and symmetric in one direction) domains when f(s)>0 for s>0 and m>2. Previously, results of this type in general bounded (and symmetric) domains had been proved only in the case 1<m<2. 相似文献
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Yasuhito Miyamoto 《Journal of Differential Equations》2010,249(8):1853-1870
Let (n?3) be a ball, and let f∈C3. We are concerned with the Neumann problem