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In this paper we study the equation −Δu+ρ−(α+2)h(ραu)=0 in a smooth bounded domain Ω where ρ(x)=dist(x,∂Ω), α>0 and h is a nondecreasing function which satisfies Keller–Osserman condition. We introduce a condition on h which implies that the equation is subcritical, i.e., the corresponding boundary value problem is well posed with respect to data given by finite measures. Under additional assumptions on h we show that this condition is necessary as well as sufficient. We also discuss b.v. problems with data given by positive unbounded measures. Our results extend results of [13] treating equations of the form −Δu+ρβuq=0 with q>1, β>−2. 相似文献
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Susana Merchán Luigi Montoro Ireneo Peral Berardino Sciunzi 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2014
In this work we deal with the existence and qualitative properties of the solutions to a supercritical problem involving the −Δp(⋅) operator and the Hardy–Leray potential. Assuming 0∈Ω, we study the regularizing effect due to the addition of a first order nonlinear term, which provides the existence of solutions with a breaking of resonance. Once we have proved the existence of a solution, we study the qualitative properties of the solutions such as regularity, monotonicity and symmetry. 相似文献
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In the well-known work of P.-L. Lions [The concentration–compactness principle in the calculus of variations, The locally compact case, part 1. Ann. Inst. H. Poincaré, Analyse Non Linéaire 1 (1984) 109–1453] existence of positive solutions to the equation -Δu+u=b(x)up-1, u>0, u∈H1(RN), p∈(2,2N/(N-2)) was proved under assumption b(x)?b∞?lim|x|→∞b(x). In this paper we prove the existence for certain functions b satisfying the reverse inequality b(x)<b∞. For any periodic lattice L in RN and for any b∈C(RN) satisfying b(x)<b∞, b∞>0, there is a finite set Y⊂L and a convex combination bY of b(·-y), y∈Y, such that the problem -Δu+u=bY(x)up-1 has a positive solution u∈H1(RN). 相似文献
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Let r,s∈]1,2[ and λ,μ∈]0,+∞[. In this paper, we deal with the existence and multiplicity of nonnegative and nonzero solutions of the Dirichlet problem with 0 boundary data for the semilinear elliptic equation −Δu=λus−1−ur−1 in Ω⊂RN, where N≥2. We prove that there exists a positive constant Λ such that the above problem has at least two solutions, at least one solution or no solution according to whether λ>Λ, λ=Λ or λ<Λ. In particular, a result by Hernandéz, Macebo and Vega is improved and, for the semilinear case, a result by Díaz and Hernandéz is partially extended to higher dimensions. Finally, an answer to a conjecture, recently stated by the author, is also given. 相似文献
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In this article we give the existence results for the quasilinear elliptic problem Δpu=b(x)f(u) in Ω,u(x)→∞ as x→∂Ω, where Ω⊆RN is a domain. 相似文献
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In this paper, we consider the problem (Pε) : Δ2u=un+4/n-4+εu,u>0 in Ω,u=Δu=0 on ∂Ω, where Ω is a bounded and smooth domain in Rn,n>8 and ε>0. We analyze the asymptotic behavior of solutions of (Pε) which are minimizing for the Sobolev inequality as ε→0 and we prove existence of solutions to (Pε) which blow up and concentrate around a critical point of the Robin's function. Finally, we show that for ε small, (Pε) has at least as many solutions as the Ljusternik–Schnirelman category of Ω. 相似文献