arXiv subjects

Qihu Zhang

Publications and source records attributed to Qihu Zhang.

4 recordsLinked to original sources

arXiv2020

Multiple solutions of double phase variational problems with variable exponent

This paper deals with the existence of multiple solutions for the quasilinear equation $-\mathrm{div}\,\mathbf{A}(x,\nabla u)| u| ^{α(x)-2}u=f(x,u)$ in $ \mathbb{R} ^{N}$, which involves a general variable exponent elliptic operator $\mathbf{ A}$ in divergence form. The problem corresponds to double phase anisotropic phenomena, in the sense that the differential operator has behaviors like $ | ξ| ^{q(x)-2}ξ$ for small $| ξ| $ and like $| ξ| ^{p(x)-2}ξ$ for large $ | ξ| $, where $1<α(\cdot )\leq p(\cdot )<q(\cdot )<N$. Our aim is to approach variationally the problem by using the tools of critical points theory in generalized Orlicz-Sobolev spaces with variable exponent. Our results extend the previous works Azzollini, d'Avenia, and Pomponio (2014) and Chorfi and Rădulescu (2016), from the case when exponents $p$ and $q$ are constant, to the case when $p(\cdot )$ and $% q(\cdot )$ are functions. We also substantially weaken some of their hypotheses overcome the lack of compactness by using the weighting method.

math.AP↗

arXiv2018

Nonhomogeneous Dirichlet problems without the Ambrosetti-Rabinowitz condition

We consider the existence of solutions of the following $p(x)$-Laplacian Dirichlet problem without the Ambrosetti-Rabinowitz condition: $-\mbox{div}(|\nabla u|^{p(x)-2}\nabla u)=f(x,u) \text{ in }Ω,$ and $u=0,\text{ on }\partial Ω.$ We give a new growth condition and we point out its importance for checking the Cerami compactness condition. We prove the existence of solutions of the above problem via the critical point theory, and also provide some multiplicity properties. Our results extend previous work by Q. Zhang and C. Zhao, Existence of strong solutions of a $p(x)$-Laplacian Dirichlet problem without the Ambrosetti-Rabinowitz condition, Comp. Math. Appl. 69 (2015), 1-12, and we establish the existence of solutions under weaker hypotheses on the nonlinear term.

math.AP↗

arXiv2016

Multiplicity of strong solutions for a class of elliptic problems without the Ambrosetti-Rabinowitz condition in $\mathbb{R}^{N}$

We investigate the existence and multiplicity of solutions to the following $p(x)$-Laplacian problem in $\mathbb{R}^{N}$ via critical point theory \begin{equation*} \left\{ \begin{array}{l} -\bigtriangleup _{p(x)}u+V(x)\left\vert u\right\vert ^{p(x)-2}u=f(x,u),\text{ in } \mathbb{R}^{N}, \\ u\in W^{1,p(\cdot )}(\mathbb{R}^{N}). \end{array} \right. \end{equation*} We propose a new set of growth conditions which matches the variable exponent nature of the problem. Under this new set of assumptions, we manage to verify the Cerami compactness condition. Therefore, we succeed in proving the existence of multiple solutions to the above problem without the well-known Ambrosetti--Rabinowitz type growth condition. Meanwhile, we could also characterize the pointwise asymptotic behaviors of these solutions. In our main argument, the idea of localization, decomposition of the domain, regularity of weak solutions and comparison principle are crucial ingredients among others.

math.AP↗

arXiv2016

Multiple solutions with constant sign of a Dirichlet problem for a class of elliptic systems with variable exponent growth

We investigate the following Dirichlet problem with variable exponents: \begin{equation*} \left\{ \begin{array}{l} -\bigtriangleup _{p(x)}u=λα(x)\left\vert u\right\vert ^{α(x)-2}u\left\vert v\right\vert ^{β(x)}+F_{u}(x,u,v),\text{ in }Ω, \\ -\bigtriangleup _{q(x)}v=λβ(x)\left\vert u\right\vert ^{α(x)}\left\vert v\right\vert ^{β(x)-2}v+F_{v}(x,u,v),\text{ in }Ω, \\ u=0=v,\text{ on }\partial Ω. \end{array} \right. \end{equation*} We present here, in the system setting, a new set of growth conditions under which we manage to use a novel method to verify the Cerami compactness condition. By localization argument, decomposition technique and variational methods, we are able to show the existence of multiple solutions with constant sign for the problem without the well-known Ambrosetti--Rabinowitz type growth condition. More precisely, we manage to show that the problem admits four, six and infinitely many solutions respectively.

math.AP↗