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In this paper, we use new analyses to assert that there are three positive solutions of Eq. (1.1) in infinite cylinder domain with hole . 相似文献
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Jiqin Deng 《Journal of Mathematical Analysis and Applications》2007,336(2):1395-1405
In this paper, by using the fixed point theory, under quite general conditions on the nonlinear term, we obtain an existence result of bounded positive solutions of semilinear elliptic equations in exterior domain of Rn, n?3. 相似文献
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In this paper, we study the effect of domain shape on the number of positive and nodal (sign-changing) solutions for a class of semilinear elliptic equations. We prove a semilinear elliptic equation in a domain Ω that contains m disjoint large enough balls has m2 2-nodal solutions and m positive solutions. 相似文献
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Benlong Xu 《Journal of Mathematical Analysis and Applications》2008,341(2):783-790
In this paper, we establish an exact multiplicity result of solutions for a class of semilinear elliptic equation. We also obtain a precise global bifurcation diagram of the solution set. As a result, an open problem presented by C.-H. Hsu and Y.-W. Shih [C.-H. Hsu, Y.-W. Shih, Solutions of semilinear elliptic equations with asymptotic linear nonlinearity, Nonlinear Anal. 50 (2002) 275-283] is completely solved. Our argument is mainly based on bifurcation theory and continuation method. 相似文献
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Francesca Gladiali 《Journal of Mathematical Analysis and Applications》2010,369(1):306-311
We consider the problem
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We consider the semilinear elliptic equation Δu+K(|x|)up=0 in RN for N>2 and p>1, and study separation phenomena of positive radial solutions. With respect to intersection and separation, we establish a classification of the solution structures, and investigate the structures of intersection, partial separation and separation. As a consequence, we obtain the existence of positive solutions with slow decay when the oscillation of the function r−?K(r) with ?>−2 around a positive constant is small near r=∞ and p is sufficiently large. Moreover, if the assumptions hold in the whole space, the equation has the structure of separation and possesses a singular solution as the upper limit of regular solutions. We also reveal that the equation changes its nature drastically across a critical exponent pc which is determined by N and the order of the behavior of K(r) as r=|x|→0 and ∞. In order to understand how subtle the structure is on K at p=pc, we explain the criticality in a similar way as done by Ding and Ni (1985) [6] for the critical Sobolev exponent p=(N+2)/(N−2). 相似文献
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Tsing-San Hsu 《Journal of Mathematical Analysis and Applications》2007,332(2):814-832
In this paper, assume that h is nonnegative and ‖hL2‖>0, we prove that if ‖hL2‖ is sufficiently small, then there are at least three positive solutions of Eq. (1) in an exterior cylinder domain. 相似文献
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Soohyun Bae 《Journal of Differential Equations》2009,247(5):1616-1635
We establish that the elliptic equation Δu+K(x)up+μf(x)=0 in Rn has a continuum of positive entire solutions for small μ?0 under suitable conditions on K, p and f. In particular, K behaves like l|x| at ∞ for some l?−2, but may change sign in a compact region. For given l>−2, there is a critical exponent pc=pc(n,l)>1 in the sense that the result holds for p?pc and involves partial separation of entire solutions. The partial separation means that the set of entire solutions possesses a non-trivial subset in which any two solutions do not intersect. The observation is well known when K is non-negative. The point of the paper is to remove the sign condition on compact region. When l=−2, the result holds for any p>1 while pc is decreasing to 1 as l decreases to −2. 相似文献
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Existence and uniqueness of large positive solutions are obtained for some semilinear elliptic equations with critical and supercritical growth on general bounded smooth domains. It is shown that the large positive solution develops a boundary layer. The boundary derivative estimate of the large solution is also established. 相似文献
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We consider a special class of radial solutions of semilinear equations −Δu=g(u) in the unit ball of Rn. It is the class of semi-stable solutions, which includes local minimizers, minimal solutions, and extremal solutions. We establish sharp pointwise, Lq, and Wk,q estimates for semi-stable radial solutions. Our regularity results do not depend on the specific nonlinearity g. Among other results, we prove that every semi-stable radial weak solution is bounded if n?9 (for every g), and belongs to H3=W3,2 in all dimensions n (for every g increasing and convex). The optimal regularity results are strongly related to an explicit exponent which is larger than the critical Sobolev exponent. 相似文献
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Tsung-fang Wu 《Journal of Mathematical Analysis and Applications》2007,325(2):1280-1294
In this paper, we study the effect of domain shape on the multiplicity of positive solutions for the semilinear elliptic equations. We prove a Palais-Smale condition in unbounded domains and assert that the semilinear elliptic equation in unbounded domains has multiple positive solutions. 相似文献
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This paper is devoted to the study of positive solutions of the semilinear elliptic equation Δu+K(|x|)u−p=0, x∈Rn with n?3 and p>0. Asymptotic behaviours of sky states and uniqueness of singular sky states are obtained via invariant manifold theory of dynamical systems. The Dirichlet problem in exterior domains is also studied. It is proved that this problem has infinitely many positive solutions with fast growth. 相似文献
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Filomena Pacella P.N. Srikanth 《Journal of Mathematical Analysis and Applications》2008,341(1):131-139
In this paper we consider the problem
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Multiplicity results for semilinear elliptic equations are obtained under one-sided growth conditions on the nonlinearity. Techniques of nonsmooth critical point theory are employed. 相似文献
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Soohyun Bae 《Journal of Differential Equations》2007,237(1):159-197
We establish that for n?3 and p>1, the elliptic equation Δu+K(x)up=0 in Rn possesses a continuum of positive entire solutions with logarithmic decay at ∞, provided that a locally Hölder continuous function K?0 in Rn?{0}, satisfies K(x)=O(σ|x|) at x=0 for some σ>−2, and 2|x|K(x)=c+O([log|x|]−θ) near ∞ for some constants c>0 and θ>1. The continuum contains at least countably many solutions among which any two do not intersect. This is an affirmative answer to an open question raised in [S. Bae, T.K. Chang, On a class of semilinear elliptic equations in Rn, J. Differential Equations 185 (2002) 225-250]. The crucial observation is that in the radial case of K(r)=K(|x|), two fundamental weights, and , appear in analyzing the asymptotic behavior of solutions. 相似文献
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Tsung-Fang Wu 《Journal of Mathematical Analysis and Applications》2006,318(1):253-270
In this paper, we study the combined effect of concave and convex nonlinearities on the number of positive solutions for semilinear elliptic equations with a sign-changing weight function. With the help of the Nehari manifold, we prove that there are at least two positive solutions for Eq. (Eλ,f) in bounded domains. 相似文献