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Register a new userNonlocal problems with critical Hardy nonlinearity
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By means of variational methods we establish existence and multiplicity of solutions for a class of nonlinear nonlocal problems involving the fractional p-Laplacian and a combined Sobolev and Hardy nonlinearity at subcritical and critical growth.
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We study a class of elliptic problems with homogeneous Dirichlet boundary condition and a nonlinear reaction term $f$ which is nonlocal depending on the $L^{p}$-norm of the unknown function. The nonlinearity $f$ can make the problem degenerate since it may even have multiple singularities in the nonlocal variable. We use fixed point arguments for an appropriately defined solution map to produce multiplicity of classical positive solutions with ordered norms.
Let $Omega subset {mathbb R}^N$ ($N geq 3$) be a $C^2$ bounded domain and $delta$ be the distance to $partial Omega$. We study positive solutions of equation (E) $-L_mu u+ g(| abla u|) = 0$ in $Omega$ where $L_mu=Delta + frac{mu}{delta^2} $, $mu in (0,frac{1}{4}]$ and $g$ is a continuous, nondecreasing function on ${mathbb R}_+$. We prove that if $g$ satisfies a singular integral condition then there exists a unique solution of (E) with a prescribed boundary datum $ u$. When $g(t)=t^q$ with $q in (1,2)$, we show that equation (E) admits a critical exponent $q_mu$ (depending only on $N$ and $mu$). In the subcritical case, namely $1<q<q_mu$, we establish some a priori estimates and provide a description of solutions with an isolated singularity on $partial Omega$. In the supercritical case, i.e. $q_muleq q<2$, we demonstrate a removability result in terms of Bessel capacities.
In this paper we prove the existence of the global attractor for the wave equation with nonlocal weak damping, nonlocal anti-damping and critical nonlinearity.
In this paper we study the positive solutions of sub linear elliptic equations with a Hardy potential which is singular at the boundary. By means of ODE techniques a fairly complete picture of the class of radial solutions is given. Local solutions with a prescribed growth at the boundary are constructed by means of contraction operators. Some of those radial solutions are then used to construct ordered upper and lower solutions in general domains. By standard iteration arguments the existence of positive solutions is proved. An important tool is the Hardy constant.
In this paper, we first establish a criterion based on contractive function for the existence of polynomial attractors. This criterion only involves some rather weak compactness associated with the repeated limit inferior and requires no compactness, which makes it suitable for critical cases. Then by this abstract theorem, we verify the existence of a polynomial attractor and estimate its attractive speed for the wave equation with nonlocal weak damping, anti-damping and critical nonlinearity.
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