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We prove the existence of solutions of nonlinear elliptic equations with first-order terms having “natural growth” with respect
to the gradient. The assumptions on the source terms lead to the existence of possibly unbounded solutions (though with exponential
integrability). The domain Ω is allowed to have infinite Lebesgue measure.
Received: April 13, 2001; in final form: September 29, 2001?Published online: July 9, 2002 相似文献
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The aim of this paper is to study the qualitative behavior of large solutions to the following problem
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Zhijun Zhang 《Nonlinear Analysis: Theory, Methods & Applications》2010,73(10):3348-3363
In this paper, we study the boundary behavior of solutions to boundary blow-up elliptic problems , where Ω is a bounded domain with smooth boundary in RN, q>0, , which is positive in Ω and may be vanishing on the boundary and rapidly varying near the boundary, and f is rapidly varying or normalized regularly varying at infinity. 相似文献
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In this note, we investigate the regularity of the extremal solution u? for the semilinear elliptic equation −△u+c(x)⋅∇u=λf(u) on a bounded smooth domain of Rn with Dirichlet boundary condition. Here f is a positive nondecreasing convex function, exploding at a finite value a∈(0,∞). We show that the extremal solution is regular in the low-dimensional case. In particular, we prove that for the radial case, all extremal solutions are regular in dimension two. 相似文献
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Cyril Imbert 《Journal of Differential Equations》2011,250(3):1553-1574
In this paper, we study fully non-linear elliptic equations in non-divergence form which can be degenerate or singular when “the gradient is small”. Typical examples are either equations involving the m-Laplace operator or Bellman-Isaacs equations from stochastic control problems. We establish an Alexandroff-Bakelman-Pucci estimate and we prove a Harnack inequality for viscosity solutions of such non-linear elliptic equations. 相似文献
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Stanley Alama Gabriella Tarantello 《Calculus of Variations and Partial Differential Equations》1993,1(4):439-475
This paper concerns semilinear elliptic equations whose nonlinear term has the formW(x)f(u) whereW changes sign. We study the existence of positive solutions and their multiplicity. The important role played by the negative part ofW is contained in a condition which is shown to be necessary for homogeneousf. More general existence questions are also discussed.Supported in part by NSF grant DMS9003149. 相似文献
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We consider the Dirichlet problem for a class of anisotropic degenerate elliptic equations. 相似文献
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Reika Fukuizumi Tohru Ozawa 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2005,56(6):1000-1011
Exponential decay estimates are obtained for complex-valued solutions to nonlinear elliptic equations in
where the linear term is given by Schr?dinger operators H = − Δ + V with nonnegative potentials V and the nonlinear term is given by a single power with subcritical Sobolev exponent in the attractive case. We describe specific
rates of decay in terms of V, some of which are shown to be optimal. Moreover, our estimates provide a unified understanding of two distinct cases in
the available literature, namely, the vanishing potential case V = 0 and the harmonic potential case V(x) = |x|2.
Dedicated to Professor Jun Uchiyama on the occasion of his sixtieth birthday
Received: May 4, 2004 相似文献
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M. Badiale 《Nonlinear Analysis: Theory, Methods & Applications》2010,72(2):602-617
We prove the existence of nonnegative symmetric solutions to the semilinear elliptic equation
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We investigate the homogeneous Dirichlet boundary value problem for a class of second-order nonlinear elliptic partial differential equations with a gradient term and singular data. Under general conditions on the data, we study the behaviour of the solution near the boundary of the domain. Under suitable additional conditions we also investigate the second-order term in the asymptotic expansion of the solution in terms of the distance from the boundary. 相似文献
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In this paper, we study the existence of infinitely many nontrivial solutions for a class of semilinear elliptic equations −△u+a(x)u=g(x,u) in a bounded smooth domain of RN(N≥3) with the Dirichlet boundary value, where the primitive of the nonlinearity g is of superquadratic growth near infinity in u and the potential a is allowed to be sign-changing. Recent results in the literature are generalized and significantly improved. 相似文献
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Fernando Charro Eduardo Colorado Ireneo Peral 《Journal of Differential Equations》2009,246(11):4221-1579
We deal with existence, non-existence and multiplicity of solutions to the model problem
(P) 相似文献
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T. del Vecchio 《Potential Analysis》1995,4(2):185-203
In this paper we prove the existence of solutions of nonlinear equations of the type-div(a(x, u, Du)+H(x, u, Du)=f, wherea andH are Caratheodory functions andf is a bounded Radon measure. We remark that the operator can be not coercive. We give also some regularity results. 相似文献
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We consider the Dirichlet problem for a class of anisotropic degenerate elliptic equations. New a priori estimates for solutions and for the gradient of solutions are established. Based on these estimates sufficient conditions guaranteeing the solvability of the problem are formulated. The results are new even in the semilinear case when the principal part is the Laplace operator. 相似文献
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Lucas C.F. Ferreira 《Journal of Differential Equations》2011,250(4):2045-2063
We study the equation Δu+u|u|p−1+V(x)u+f(x)=0 in Rn, where n?3 and p>n/(n−2). The forcing term f and the potential V can be singular at zero, change sign and decay polynomially at infinity. We can consider anisotropic potentials of form h(x)|x|−2 where h is not purely angular. We obtain solutions u which blow up at the origin and do not belong to any Lebesgue space Lr. Also, u is positive and radial, in case f and V are. Asymptotic stability properties of solutions, their behavior near the singularity, and decay are addressed. 相似文献
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Guozhen Lu Peiyong Wang Jiuyi Zhu 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2012
Liouville-type theorems are powerful tools in partial differential equations. Boundedness assumptions of solutions are often imposed in deriving such Liouville-type theorems. In this paper, we establish some Liouville-type theorems without the boundedness assumption of nonnegative solutions to certain classes of elliptic equations and systems. Using a rescaling technique and doubling lemma developed recently in Polá?ik et al. (2007) [20], we improve several Liouville-type theorems in higher order elliptic equations, some semilinear equations and elliptic systems. More specifically, we remove the boundedness assumption of the solutions which is required in the proofs of the corresponding Liouville-type theorems in the recent literature. Moreover, we also investigate the singularity and decay estimates of higher order elliptic equations. 相似文献
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We propose results on interior Morrey, BMO and H?lder regularity for the strong solutions to linear elliptic systems of order 2b with discontinuous coefficients and right-hand sides belonging to the Morrey space Lp,λ.
Received: 20 October 2004 相似文献