We consider a double phase problem driven by the sum of the p-Laplace operator and a weighted q-Laplacian ($$q<p$$), with a weight function which is not bounded away from zero. The reaction term is $$(p-1)$$-superlinear. Employing the Nehari method, we show that the equation has a ground state solution of constant sign and a nodal (sign-changing) solution.

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Ground state and nodal solutions for a class of double phase problems

Double phase image restoration

We consider a double phase problem in RN with a q−1-superlinear subcritical Caratheodory reaction term, which does not satisfy the Ambrosetti–Rabinowitz condition. Using variational methods based on the topological degree and critical point theory together with the Nehari manifold method and deformation lemma, we find three nontrivial solutions: among them, the first one is the positive ground state solution, the second one is the negative ground state solution, and the third one is the least energy sign-changing solution, which changes sign only once. We also discuss the existence of infinitely many solutions for even energy functional and that of radial solutions.

Multiplicity results for double phase problems in RN

We consider a parametric double phase problem with Robin boundary condition. We prove two existence theorems. In the first the reaction is ( p − 1 )-superlinear and the solutions produced are asymptotically big as λ → 0 + . In the second the conditions on the reaction are essentially local at zero and the solutions produced are asymptotically small as λ → 0 + .

/pdf/solutions-for-parametric-double-phase-robin-problems-20554lxgxc.pdf

Solutions for parametric double phase Robin problems

In this paper, we deal with the following double phase problem $$ \left\{\begin{array}{ll} -\mbox{div}\left(|\nabla u|^{p-2}\nabla u+a(x)|\nabla u|^{q-2}\nabla u\right)= \gamma\left(\displaystyle\frac{|u|^{p-2}u}{|x|^p}+a(x)\displaystyle\frac{|u|^{q-2}u}{|x|^q}\right)+f(x,u) & \mbox{in } \Omega,\\ u=0 & \mbox{in } \partial\Omega, \end{array} \right. $$ where $\Omega\subset\mathbb R^N$ is an open, bounded set with Lipschitz boundary, $0\in\Omega$, $N\geq2$, $1<p<q<N$, weight $a(\cdot)\geq0$, $\gamma$ is a real parameter and $f$ is a subcritical function. By variational method, we provide the existence of a non-trivial weak solution on the Musielak-Orlicz-Sobolev space $W^{1,\mathcal H}_0(\Omega)$, with modular function $\mathcal H(t,x)=t^p+a(x)t^q$. For this, we first introduce the Hardy inequalities for space $W^{1,\mathcal H}_0(\Omega)$, under suitable assumptions on $a(\cdot)$.

A double phase problem involving Hardy potentials

In the present paper, in view of the variational approach, we consider the existence and multiplicity of weak solutions for a class of the double phase problem − div ( | ∇ u | p − 2 ∇ u + a ( x ) | ∇ u | q − 2 ∇ u ) = λ f ( x , u ) , in Ω , u = 0 , on ∂ Ω , where N ≥ 2 and 1 p q N . Firstly, by the Fountain and Dual Theorem with Cerami condition, we obtain some existence of infinitely many solutions for the above problem under some weaker assumptions on f . Secondly, we prove that this problem has at least one nontrivial solution for any parameter λ > 0 small enough, and also that the solution blows up, in the Sobolev norm, as λ → 0 + . Finally, by imposing additional assumptions on f , we establish the existence of infinitely many solutions by using Krasnoselskii’s genus theory for the above equation.

Multiple solutions for a class of double phase problem without the Ambrosetti–Rabinowitz conditions

We study the following double-phase problem −div(|∇u|p−2∇u+a(x)|∇u|q−2∇u)=λf(x,u),in Ω,u=0,on ∂Ω, where N≥ 2 and 1<p<q<N. The aim is to determine the precise positive interval of λ for which the pr...

On a class of double-phase problem without Ambrosetti–Rabinowitz-type conditions

On finite generalized quadrangles with rmPSL(2, q) as an automorphism group

In this paper, we study the p(x)$p(x)$ ‐curl problem of the type: ∇×(|∇×u|p(x)−2∇×u)+a(x)|u|p(x)−2u=λf(x,u),inΩ,∇·u=0,inΩ,|∇×u|p(x)−2∇×u×n=0,u·n=0,on∂Ω,\begin{equation*} \hspace*{13pc}{\left\lbrace \def\eqcellsep{&}\begin{array}{ll}\nabla \times \big (|\nabla \times \mathbf {u} |^{p(x)-2}\nabla \times \mathbf {u}\big )+a(x)|\mathbf {u}|^{p(x)-2}\mathbf {u} =\lambda \mathbf {f}(x,\mathbf {u}), &{\rm in}\; \Omega , \\ [0.1cm] \nabla \cdot \mathbf {u}=0,\;&{\rm in}\; \Omega , \\ [0.1cm] |\nabla \times \mathbf {u}|^{p(x)-2}\nabla \times \mathbf {u} \times \mathbf {n}=0,\mathbf {u}\cdot \mathbf {n}=0, \; &\text{on} \; \partial \Omega , \end{array} \right.} \end{equation*}where Ω⊂R3$\Omega \subset \mathbb {R}^{3}$ is a bounded simply connected domain with a C1, 1 boundary denoted by ∂Ω$\partial \Omega$ , λ>0$\lambda >0$ , p(x):Ω¯→(1,+∞)$p(x):\overline{\Omega }\rightarrow (1,+\infty )$ is a continuous function, a(x)∈L∞(Ω)$a(x)\in L^{\infty }(\Omega )$ , and f:Ω¯×R3→R3$\mathbf {f}:\overline{\Omega }\times \mathbb {R}^{3}\rightarrow \mathbb {R}^{3}$ is a Carathéodory function. We obtain the existence and multiplicity of solutions for a class of p(x)$p(x)$ ‐curl systems in the absence of Ambrosetti–Rabinowitz condition under superlinear case. Besides, we also obtain infinitely many solutions under sublinear case.

Multiple solutions for p(x)${p(x)}$ ‐curl systems with nonlinear boundary condition

On finite generalized quadrangles with 
            
              
            
            $$\textrm{PSL}(2,q)$$
            
              
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           as an au

Jian-Fang Lu

Papers

Multiple solutions for a class of double phase problem without the Ambrosetti–Rabinowitz conditions

On a class of double-phase problem without Ambrosetti–Rabinowitz-type conditions

On finite generalized quadrangles with rmPSL(2, q) as an automorphism group

Multiple solutions for p(x)${p(x)}$ ‐curl systems with nonlinear boundary condition

On finite generalized quadrangles with $$\textrm{PSL}(2,q)$$ PSL ( 2 , q ) as an au