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Register a new userOn Strongly Nonlinear Eigenvalue Problems in the Framework of Nonreflexive Orlicz-Sobolev Spaces
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It is established existence and multiplicity of solutions for strongly nonlinear problems driven by the $Phi$-Laplacian operator on bounded domains. Our main results are stated without the so called $Delta_{2}$ condition at infinity which means that the underlying Orlicz-Sobolev spaces are not reflexive.
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In this paper we are concerned with some abstract results regarding to fractional Orlicz-Sobolev spaces. Precisely, we ensure the compactness embedding for the weighted fractional Orlicz-Sobolev space into the Orlicz spaces, provided the weight is unbounded. We also obtain a version of Lions vanishing Lemma for fractional Orlicz-Sobolev spaces, by introducing new techniques to overcome the lack of a suitable interpolation law. Finally, as a product of the abstract results, we use a minimization method over the Nehari manifold to prove the existence of ground state solutions for a class of nonlinear Schr{o}dinger equations, taking into account unbounded or bounded potentials.
We prove the solvability in Sobolev spaces of the conormal derivative problem for the stationary Stokes system with irregular coefficients on bounded Reifenberg flat domains. The coefficients are assumed to be merely measurable in one direction, which may differ depending on the local coordinate systems, and have small mean oscillations in the other directions. In the course of the proof, we use a local version of the Poincare inequality on Reifenberg flat domains, the proof of which is of independent interest.
We study the non-existence, existence and multiplicity of positive solutions to the following nonlinear Kirchhoff equation:% begin{equation*} left{ begin{array}{l} -Mleft( int_{mathbb{R}^{3}}leftvert abla urightvert ^{2}dxright) Delta u+mu Vleft( xright) u=Q(x)leftvert urightvert ^{p-2}u+lambda fleft( xright) utext{ in }mathbb{R}^{N}, uin H^{1}left( mathbb{R}^{N}right) ,% end{array}% right. end{equation*}% where $Ngeq 3,2<p<2^{ast }:=frac{2N}{N-2},Mleft( tright) =at+b$ $left( a,b>0right) ,$ the potential $V$ is a nonnegative function in $mathbb{R}% ^{N}$ and the weight function $Qin L^{infty }left( mathbb{R}^{N}right) $ with changes sign in $overline{Omega }:=left{ V=0right} .$ We mainly prove the existence of at least two positive solutions in the cases that $% left( iright) $ $2<p<min left{ 4,2^{ast }right} $ and $0<lambda <% left[ 1-2left[ left( 4-pright) /4right] ^{2/p}right] lambda _{1}left( f_{Omega }right) ;$ $left( iiright) $ $pgeq 4,lambda geq lambda _{1}left( f_{Omega }right) $ and near $lambda _{1}left( f_{Omega }right) $ for $mu >0$ sufficiently large, where $lambda _{1}left( f_{Omega }right) $ is the first eigenvalue of $-Delta $ in $% H_{0}^{1}left( Omega right) $ with weight function $f_{Omega }:=f|_{% overline{Omega }},$ whose corresponding positive principal eigenfunction is denoted by $phi _{1}.$ Furthermore, we also investigated the non-existence and existence of positive solutions if $a,lambda $ belongs to different intervals.
We construct an efficient approach to deal with the global regularity estimates for a class of elliptic double-obstacle problems in Lorentz and Orlicz spaces. The motivation of this paper comes from the study on an abstract result in the viewpoint of the fractional maximal distributions and this work also extends some regularity results proved in cite{PN_dist} by using the weighted fractional maximal distributions (WFMDs). We further investigate a pointwise estimates of the gradient of weak solutions via fractional maximal operators and Riesz potential of data. Moreover, in the setting of the paper, we are led to the study of problems with nonlinearity is supposed to be partially weak BMO condition (is measurable in one fixed variable and only satisfies locally small-BMO seminorms in the remaining variables).
This paper is concerned with the development and analysis of an iterative solver for high-dimensional second-order elliptic problems based on subspace-based low-rank tensor formats. Both the subspaces giving rise to low-rank approximations and corresponding sparse approximations of lower-dimensional tensor components are determined adaptively. A principal obstruction to a simultaneous control of rank growth and accuracy turns out to be the fact that the underlying elliptic operator is an isomorphism only between spaces that are not endowed with cross norms. Therefore, as central part of this scheme, we devise a method for preconditioning low-rank tensor representations of operators. Under standard assumptions on the data, we establish convergence to the solution of the continuous problem with a guaranteed error reduction. Moreover, for the case that the solution exhibits a certain low-rank structure and representation sparsity, we derive bounds on the computational complexity, including in particular bounds on the tensor ranks that can arise during the iteration. We emphasize that such assumptions on the solution do not enter in the formulation of the scheme, which in fact is shown to detect them automatically. Our findings are illustrated by numerical experiments that demonstrate the practical efficiency of the method in high spatial dimensions.
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