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Register a new userNodal solutions to a Neumann problem for a class of (p_1,p_2)-Laplacian systems
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Nodal solutions of a parametric (p_1,p_2)-Laplacian system, with Neumann boundary conditions, are obtained by chiefly constructing appropriate sub-super-solution pairs.
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We study the regularity up to the boundary of solutions to the Neumann problem for the fractional Laplacian. We prove that if $u$ is a weak solution of $(-Delta)^s u=f$ in $Omega$, $mathcal N_s u=0$ in $Omega^c$, then $u$ is $C^alpha$ up tp the boundary for some $alpha>0$. Moreover, in case $s>frac12$, we then show that $uin C^{2s-1+alpha}(overlineOmega)$. To prove these results we need, among other things, a delicate Moser iteration on the boundary with some logarithmic corrections. Our methods allow us to treat as well the Neumann problem for the regional fractional Laplacian, and we establish the same boundary regularity result. Prior to our results, the interior regularity for these Neumann problems was well understood, but near the boundary even the continuity of solutions was open.
In this paper, we are concern with the multiplicity of solutions for a p-Laplacian problem. A weaker super-quadratic assumptions is required on the nonlinearity. Under the weaker condition we give a new proof for the infinite solutions having a prescribed number of nodes to the problem. It turns out that the weaker condition on nonlinearity suffices to guarantee the infinitely many solutions. At the same time, a global characterization of the critical values of the nodal radial solutions are given.
In this paper, we consider the existence of nodal solutions with two bubbles to the slightly subcritical problem with the fractional Laplacian begin{equation*} left{aligned &(-Delta)^su=|u|^{p-1-varepsilon}u mbox{in} Omega &u=0 mbox{on} partialOmega, endaligned right. end{equation*} where $Omega$ is a smooth bounded domain in $mathbb R^N$, $N>2s$, $0<s<1$, $ p=frac{N+2s}{N-2s}$ and $varepsilon>0$ is a small parameter, which can be seen as a nonlocal analog of the results of Bartsch, Micheletti and Pistoia (2006) cite{Bartsch1}.
We establish the existence and multiplicity of positive solutions to the problems involving the fractional Laplacian: begin{equation*} left{begin{array}{lll} &(-Delta)^{s}u=lambda u^{p}+f(u),,,u>0 quad &mbox{in},,Omega, &u=0quad &mbox{in},,mathbb{R}^{N}setminusOmega, end{array}right. end{equation*} where $Omegasubset mathbb{R}^{N}$ $(Ngeq 2)$ is a bounded smooth domain, $sin (0,1)$, $p>0$, $lambdain mathbb{R}$ and $(-Delta)^{s}$ stands for the fractional Laplacian. When $f$ oscillates near the origin or at infinity, via the variational argument we prove that the problem has arbitrarily many positive solutions and the number of solutions to problem is strongly influenced by $u^{p}$ and $lambda$. Moreover, various properties of the solutions are also described in $L^{infty}$- and $X^{s}_{0}(Omega)$-norms.
We prove global essential boundedness for the weak solutions of divergence form quasilinear systems. The principal part of the differential operator is componentwise coercive and supports controlled growths with respect to the solution and its gradient, while the lower order term exhibits componentwise controlled gradient growth. The x-behaviour of the nonlinearities is governed in terms of Morrey spaces.
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