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Register a new userEvolution equations with nonlocal initial conditions and superlinear growth
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We carry out an analysis of the existence of solutions for a class of nonlinear partial differential equations of parabolic type. The equation is associated to a nonlocal initial condition, written in general form which includes, as particular cases, the Cauchy multipoint problem, the weighted mean value problem and the periodic problem. The dynamic is transformed into an abstract setting and by combining an approximation technique with the Leray-Schauder continuation principle, we prove global existence results. By the compactness of the semigroup generated by the linear operator, we do not assume any Lipschitzianity, nor compactness on the nonlinear term or on the nonlocal initial condition. In addition, the exploited approximation technique coupled to a Hartman-type inequality argument, allows to treat nonlinearities with superlinear growth. Moreover, regarding the periodic case we are able to prove the existence of at least one periodic solution on the half line.
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We are concerned with a class of Kirchhoff type equations in $mathbb{R}^{N}$ as follows: begin{equation*} left{ begin{array}{ll} -Mleft( int_{mathbb{R}^{N}}| abla u|^{2}dxright) Delta u+lambda Vleft( xright) u=f(x,u) & text{in }mathbb{R}^{N}, uin H^{1}(mathbb{R}^{N}), & end{array}% right. end{equation*}% where $Ngeq 1,$ $lambda>0$ is a parameter, $M(t)=am(t)+b$ with $a,b>0$ and $min C(mathbb{R}^{+},mathbb{R}^{+})$, $Vin C(mathbb{R}^{N},mathbb{R}^{+})$ and $fin C(mathbb{R}^{N}times mathbb{R}, mathbb{R})$ satisfying $lim_{|u|rightarrow infty }f(x,u) /|u|^{k-1}=q(x)$ uniformly in $xin mathbb{R}^{N}$ for any $2<k<2^{ast}$($2^{ast}=infty$ for $N=1,2$ and $2^{ast}=2N/(N-2)$ for $Ngeq 3$). Unlike most other papers on this problem, we are more interested in the effects of the functions $m$ and $q$ on the number and behavior of solutions. By using minimax method as well as Caffarelli-Kohn-Nirenberg inequality, we obtain the existence and multiplicity of positive solutions for the above problem.
In this paper we study quasilinear elliptic equations driven by the double phase operator along with a reaction that has a singular and a parametric superlinear term and with a nonlinear Neumann boundary condition of critical growth. Based on a new equivalent norm for Musielak-Orlicz Sobolev spaces and the Nehari manifold along with the fibering method we prove the existence of at least two weak solutions provided the parameter is sufficiently small.
We give pointwise gradient bounds for solutions of (possibly non-uniformly) elliptic partial differential equations in the entire Euclidean space. The operator taken into account is very general and comprises also the singular and degenerate nonlinear case with non-standard growth conditions. The sourcing term is also allowed to have a very general form, depending on the space variables, on the solution itself, on its gradient, and possibly on higher order derivatives if additional structural conditions are satisfied.
We prove global stability results of {sl DiPerna-Lions} renormalized solutions for the initial boundary value problem associated to some kinetic equations, from which existence results classically follow. The (possibly nonlinear) boundary conditions are completely or partially diffuse, which includes the so-called Maxwell boundary conditions, and we prove that it is realized (it is not only a boundary inequality condition as it has been established in previous works). We are able to deal with Boltzmann, Vlasov-Poisson and Fokker-Planck type models. The proofs use some trace theorems of the kind previously introduced by the author for the Vlasov equations, new results concerning weak-weak convergence (the renormalized convergence and the biting $L^1$-weak convergence), as well as the Darroz`es-Guiraud information in a crucial way.
Recently, several works have been carried out in attempt to develop a theory for linear or sublinear elliptic equations involving a general class of nonlocal operators characterized by mild assumptions on the associated Green kernel. In this paper, we study the Dirichlet problem for superlinear equation (E) ${mathbb L} u = u^p +lambda mu$ in a bounded domain $Omega$ with homogeneous boundary or exterior Dirichlet condition, where $p>1$ and $lambda>0$. The operator ${mathbb L}$ belongs to a class of nonlocal operators including typical types of fractional Laplacians and the datum $mu$ is taken in the optimal weighted measure space. The interplay between the operator ${mathbb L}$, the source term $u^p$ and the datum $mu$ yields substantial difficulties and reveals the distinctive feature of the problem. We develop a new unifying technique based on a fine analysis on the Green kernel, which enables us to construct a theory for semilinear equation (E) in measure frameworks. A main thrust of the paper is to provide a fairly complete description of positive solutions to the Dirichlet problem for (E). In particular, we show that there exist a critical exponent $p^*$ and a threshold value $lambda^*$ such that the multiplicity holds for $1<p<p^*$ and $0<lambda<lambda^*$, the uniqueness holds for $1<p<p^*$ and $lambda=lambda^*$, and the nonexistence holds in other cases. Various types of nonlocal operator are discussed to exemplify the wide applicability of our theory.
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