On nonlinear evolution variational inequalities involving variable exponent

In this paper, we discuss a class of quasilinear evolution variational inequalities with variable exponent growth conditions in a generalized Sobolev space. We obtain the existence of weak solutions by means of penalty method. Moreover, we study the extinction properties of weak solutions to parabolic inequalities and provide a sufficient condition that makes the weak solutions vanish in a finite time. The existence of global attractors for weak solutions is also obtained via the theories of multi-valued semiflow

Regularity of weak solutions for nonlinear parabolic problem with p(x)p(x)-growth

In this paper, we study the nonlinear parabolic problem with $p(x)$-growth conditions in the space $W^{1,x}L^{p(x)}(Q)$, and give a regularity theorem of weak solutions for the following equation $$\frac{\partial u}{\partial t}+A(u)=0$$ where A(u)=-\mbox{div} a(x,t,u,\nabla u)+a_0(x,t,u,\nabla u), $a(x,t,u,\nabla u)$ and $a_0(x,t,u,\nabla u)$ satisfy $p(x)$-growth conditions with respect to $u$ and $\nabla u$

Existence of solutions to a class of quasilinear Schrödinger systems involving the fractional pp-Laplacian

The purpose of this paper is to investigate the existence of solutions to the following quasilinear Schr\"{o}dinger type system driven by the fractional $p$-Laplacian \begin{align*} (-\Delta)^{s}_pu+a(x)|u|^{p-2}u&=H_u(x,u,v)\quad \mbox{in } \mathbb{R}^N,\\ (-\Delta)^s_qv+b(x)|v|^{q-2}v&=H_v(x,u,v)\quad \mbox{in } \mathbb{R}^N, \end{align*} where $1<q\leq p$, $sp<N$, $(-\Delta )_m^s$ is the fractional $m$-Laplacian, the coefficients $a, b$ are two continuous and positive functions, and $H_u,H_v$ denote the partial derivatives of $H$ with respect to the second variable and the third variable. By using the mountain pass theorem, we obtain the existence of nontrivial and nonnegative solutions for the above system. The main feature of this paper is that the nonlinearities do not necessarily satisfy the Ambrosetti-Rabinowitz condition

Multiplicity of solutions to a nonlocal Choquard equation involving fractional magnetic operators and critical exponent

In this article, we study the multiplicity of solutions to a nonlocal fractional Choquard equation involving an external magnetic potential and critical exponent, namely, \displaylines{ (a+b[u]_{s,A}^2)(-\Delta)_A^su+V(x)u =\int_{\mathbb{R}^N}\frac{|u(y)|^{2_{\mu,s}^*}}{|x-y|^{\mu}}dy|u|^{2_{\mu,s}^*-2}u +\lambda h(x)|u|^{p-2}u\quad \text{in }\mathbb{R}^N, \cr [u]_{s,A}=\Big(\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^N} \frac{|u(x)-e^{i(x-y)\cdot A(\frac{x+y}{2})}u(y)|^2}{|x-y|^{N+2s}}\,dx\,dy\Big) ^{1/2} } where $a\geq 0$, b>0, $0<s<\min\{1,N/4\}$, $4s\leq \mu<N$, $V:\mathbb{R}^N\to \mathbb{R}$ is a sign-changing scalar potential, $A:\mathbb{R}^N\to \mathbb{R}^N$ is the magnetic potential, $(-\Delta )_A^s$ is the fractional magnetic operator, $\lambda>0$ is a parameter, $2_{\mu,s}^*=\frac{2N-\mu}{N-2s}$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality and $2<p<2_s^*$. Under suitable assumptions on a,b and $\lambda$, we obtain multiplicity of nontrivial solutions by using variational methods. In particular, we obtain the existence of infinitely many nontrivial solutions for the degenerate Kirchhoff case, that is, a=0, b>0

Weak solutions for nonlocal evolution variational inequalities involving gradient constraints and variable exponent

In this article, we study a class of nonlocal quasilinear parabolic variational inequality involving $p(x)$-Laplacian operator and gradient constraint on a bounded domain. Choosing a special penalty functional according to the gradient constraint, we transform the variational inequality to a parabolic equation. By means of Galerkin's approximation method, we obtain the existence of weak solutions for this equation, and then through a priori estimates, we obtain the weak solutions of variational inequality