2,981 research outputs found

    Caffarelli-Kohn-Nirenberg type equations of fourth order with the critical exponent and Rellich potential

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    We study the existence/nonexistence of positive solution of Δ2uμux4=uqβ2uxβin Ω, {\Delta^2u-\mu\frac{u}{|x|^4}=\frac{|u|^{q_{\beta}-2}u}{|x|^{\beta}}\quad\textrm{in $\Omega$,}} when Ω\Omega is a bounded domain and N5N\geq 5, qβ=2(Nβ)N4q_{\beta}=\frac{2(N-\beta)}{N-4}, 0β<40\leq \beta<4 and 0μ<(N(N4)4)20\leq\mu<\big(\frac{N(N-4)}{4}\big)^2. We prove the nonexistence result when Ω\Omega is an open subset of RN\mathbf R^N which is star shaped with respect to the origin. We also study the existence of positive solution in Ω\Omega when Ω\Omega is a bounded domain with non trivial topology and β=0\beta=0, μ(0,μ0)\mu\in(0,\mu_0), for certain μ0<(N(N4)4)2\mu_0<\big(\frac{N(N-4)}{4}\big)^2 and N8N\geq 8. Different behavior of PS sequences have been obtained depending on β=0\beta=0 or β>0\beta>0.Comment: 21 page

    Virtual characters on L-functions

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    In this expository note we show the inception and development of the Heilbronn characters and their application to the holomorphy of quotients of Artin L-functions. Further we use arithmetic Heilbronn characters introduced by Wong, to deal with holomorphy of quotients of certain L-functions, e.g,, L-functions associated to CM elliptic curves. Furthermore we use the supercharacter theory introduced by Diaconis and Isaacs to study Artin L-functions associated to such characters. We conclude the note surveying about various other unconditional approaches taken based on character theory of finite groups

    Infinitely many sign changing solutions of an elliptic problem involving critical Sobolev and Hardy-Sobolev exponent

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    We study the existence and multiplicity of sign changing solutions of the following equation {Δu=μu22u+u2(t)2uxt+a(x)uinΩ,u=0onΩ, \begin{cases} -\Delta u = \mu |u|^{2^{\star}-2}u+\frac{|u|^{2^{*}(t)-2}u}{|x|^t}+a(x)u \quad\text{in}\quad \Omega, u=0 \quad\text{on}\quad\partial\Omega, \end{cases} where Ω\Omega is a bounded domain in RNR^N, 0Ω0\in\partial\Omega, all the principal curvatures of Ω\partial\Omega at 00 are negative and $\mu\geq 0, \ \ a>0, \ \ N\geq 7, \ \ 0<t<2, \ \ 2^{\star}=\frac{2N}{N-2}and and 2^{\star}(t)=\frac{2(N-t)}{N-2}$

    Entire solutions for a class of elliptic equations involving pp-biharmonic operator and Rellich potentials

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    We study existence, multiplicity and qualitative properties of entire solutions for a noncompact problem related to p-biharmonic type equations with weights. More precisely, we deal with the following family of equations Δp2u=λx2pup2u+xβuq2uinRN, \Delta_{p}^2 u = \lambda|x|^{-2p}|u|^{p-2}u + |x|^{-\beta}|u|^{q-2} u\quad\text{in} \quad \mathbb R^N, where N>2pN> 2p, p>1p>1, q>pq>p, β=Nqp(N2p)\beta = N - \frac{q}{p}(N-2p) and λR\lambda\in\mathbb R is smaller than the Rellich constant.Comment: 12 page

    A note on semilinear elliptic equation with biharmonic operator and multiple critical nonlinearities

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    We study the existence and non-existence of nontrivial weak solution of Δ2uμux4=uqβ2uxβ+uq2uin RN, {\Delta^2u-\mu\frac{u}{|x|^{4}} = \frac{|u|^{q_{\beta}-2}u}{|x|^{\beta}}+|u|^{q-2}u\quad\textrm{in ${\mathbb R}^N$,}} where N5N\geq 5, qβ=2(Nβ)N4q_{\beta}=\frac{2(N-\beta)}{N-4}, 0<β<40<\beta<4, 1<q21<q\leq 2^{**} and μ<μ1:=(N(N4)4)2\mu<\mu_1:=\big(\frac{N(N-4)}{4}\big)^2. Using Pohozaev type of identity, we prove the non-existence result when 1<q<21<q< 2^{**}. On the other hand when the equation has multiple critical nonlinearities i.e. q=2q=2^{**} and (N2)2μ<μ1-(N-2)^2\leq\mu<\mu_1, we establish the existence of nontrivial solution using the Mountain-Pass theorem by Ambrosetti and Rabinowitz and the variational methods.Comment: 12 page

    Semilinear elliptic PDE's with biharmonic operator and a singular potential

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    We study the existence/nonexistence of positive solution to the problem of the type: \begin{equation}\tag{PλP_{\lambda}} \begin{cases} \Delta^2u-\mu a(x)u=f(u)+\lambda b(x)\quad\textrm{in Ω\Omega,}\\ u>0 \quad\textrm{in Ω\Omega,}\\ u=0=\Delta u \quad\textrm{on Ω\partial\Omega,} \end{cases} \end{equation} where Ω\Omega is a smooth bounded domain in RN\mathbb R^N, N5N\geq 5, a,b,fa, b, f are nonnegaive functions satisfying certain hypothesis which we will specify later. μ,λ\mu,\lambda are positive constants. Under some suitable conditions on functions a,b,fa, b, f and the constant μ\mu, we show that there exists λ>0\lambda^*>0 such that when 0<λ<λ0<\lambda<\lambda^*, (PλP_{\lambda}) admits a solution in W2,2(Ω)W01,2(Ω)W^{2,2}(\Omega)\cap W^{1,2}_0(\Omega) and for λ>λ\lambda>\lambda^*, it does not have any solution in W2,2(Ω)W01,2(Ω)W^{2,2}(\Omega)\cap W^{1,2}_0(\Omega). Moreover as λλ\lambda\uparrow\lambda^*, minimal positive solution of (PλP_{\lambda}) converges in W2,2(Ω)W01,2(Ω)W^{2,2}(\Omega)\cap W^{1,2}_0(\Omega) to a solution of (PλP_{\lambda^*}). We also prove that there exists λ~<\tilde{\lambda}^*<\infty such that λλ~\lambda^*\leq\tilde{\lambda}^* and for λ>λ~\lambda>\tilde{\lambda}^*, the above problem (PλP_{\lambda}) does not have any solution even in the distributional sense/very weak sense and there is complete {\it blow-up}. Under an additional integrability condition on bb, we establish the uniqueness of positive solution of (PλP_{\lambda^*}) in W2,2(Ω)W01,2(Ω)W^{2,2}(\Omega)\cap W^{1,2}_0(\Omega).Comment: 21 page

    Multiplicity results for (p,q)(p,\, q) fractional elliptic equations involving critical nonlinearities

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    In this paper we prove the existence of infinitely many nontrivial solutions for the class of (p,q)(p,\, q) fractional elliptic equations involving concave-critical nonlinearities in bounded domains in RN\mathbb{R}^N. Further, when the nonlinearity is of convex-critical type, we establish the multiplicity of nonnegative solutions using variational methods. In particular, we show the existence of at least catΩ(Ω)cat_{\Omega}(\Omega) nonnegative solutions.Comment: 36 page

    Multiplicity results and sign changing solutions of non-local equations with concave-convex nonlinearities

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    In this paper we prove the existence of infinitely many nontrivial solutions of the following equations driven by a nonlocal integro-differential operator LKL_K with concave-convex nonlinearities and homogeneous Dirichlet boundary conditions \begin{eqnarray*} \mathcal{L}_{K} u + \mu\, |u|^{q-1}u + \lambda\,|u|^{p-1}u &=& 0 \quad\text{in}\quad \Omega, \\[2mm] u&=&0 \quad\mbox{in}\quad\mathbb{R}^N\setminus\Omega, \end{eqnarray*} where Ω\Omega is a smooth bounded domain in RN\mathbb{R}^N, N>2sN>2s, s(0,1)s\in(0, 1), 0<q<1<pN+2sN2s0<q<1<p\leq \frac{N+2s}{N-2s}. Moreover, when LKL_K reduces to the fractional laplacian operator (Δ)s-(-\Delta)^s , p=N+2sN2sp=\frac{N+2s}{N-2s}, 12(N+2sN2s)6s\frac{1}{2}(\frac{N+2s}{N-2s})6s, λ=1\lambda=1, we find μ>0\mu^*>0 such that for any μ(0,μ)\mu\in(0,\mu^*), there exists at least one sign changing solution.Comment: 32 pages. Proof of Claim 4 in Theorem 4.1 has been modified in this versio

    Semilinear elliptic equations admitting similarity transformations

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    In this paper we study the equation Δu+ρ(α+2)h(ραu)=0-\Delta u+\rho^{-(\alpha+2)}h(\rho^{\alpha}u)=0 in a smooth bounded domain Ω\Omega where ρ(x)=dist(x,Ω)\rho(x)=\textrm{dist}\,(x,\partial \Omega), α>0\alpha>0 and hh is a non-decreasing function which satisfies Keller-Osserman condition. We introduce a condition on hh 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 hh 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 \cite{MV1} treating equations of the form Δu+ρβuq=0-\Delta u+\rho^\beta u^q=0 with q>1q>1, β>2\beta>-2.Comment: 30 page

    Sign changing solutions of p-fractional equations with concave-convex nonlinearities

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    In this article we study the existence of sign changing solution of the following p-fractional problem with concave-critical nonlinearities: \begin{eqnarray*} (-\Delta)^s_pu &=& \mu |u|^{q-1}u + |u|^{p^*_s-2}u \quad\mbox{in}\quad \Omega, u&=&0\quad\mbox{in}\quad\mathbb{R}^N\setminus\Omega, \end{eqnarray*} where s(0,1)s\in(0,1) and p2p\geq 2 are fixed parameters, 0<q<p10<q<p-1, μR+\mu\in\mathbb{R}^+ and ps=NpNpsp_s^*=\frac{Np}{N-ps}. Ω\Omega is an open, bounded domain in RN\mathbb{R}^N with smooth boundary with N>psN>ps .Comment: 28 pages. arXiv admin note: text overlap with arXiv:1603.0555
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