Abstract In this article, we consider the existence and nonexistence of positive solutions for the following critical Neumann problem: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mfenced open="{" close=""> <m:mrow> <m:mtable displaystyle="true"> <m:mtr> <m:mtd columnalign="left"> <m:mo>−</m:mo> <m:msub> <m:mrow> <m:mi>Δ</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msub> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mi>λ</m:mi> <m:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mo>=</m:mo> <m:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mo>*</m:mo> </m:mrow> </m:msup> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mo>+</m:mo> <m:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>,</m:mo> </m:mtd> <m:mtd columnalign="left"> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>;</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="left"> <m:msup> <m:mrow> <m:mo>∣</m:mo> <m:mrow> <m:mo>∇</m:mo> </m:mrow> <m:mi>u</m:mi> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo>∇</m:mo> </m:mrow> <m:mi>u</m:mi> <m:mo>⋅</m:mo> <m:mi>ν</m:mi> <m:mo>=</m:mo> <m:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mi>♯</m:mi> </m:mrow> </m:msup> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mo>,</m:mo> </m:mtd> <m:mtd columnalign="left"> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:mo>∂</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>,</m:mo> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:math> \left\{\begin{array}{ll}-{\Delta }_{p}u+\lambda {u}^{p-1}={u}^{{p}^{* }-1}+f\left(x,u),&amp; x\in \Omega ;\\ {| \nabla u| }^{p-2}\nabla u\cdot \nu ={u}^{{p}^{\sharp }-1},&amp; x\in \partial \Omega ,\end{array}\right. where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>⊂</m:mo> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:math> \Omega \subset {{\mathbb{R}}}^{n} is a bounded domain with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>C</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> {C}^{1} boundary, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>3</m:mn> </m:math> n\ge 3 , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mo>*</m:mo> </m:m

Abstract In this article, we propose a model describing laminar shallow water flows with two velocities to handle an uneven bottom. The model is established by taking the different velocities into account. The source terms generated from the bottom topography prevent us from solving the Riemann problem directly. We first derive the elementary waves including the stationary wave where the global entropy condition is used to ensure uniqueness. Then, we analyze the resonance phenomenon and coalescence of waves by classifying the initial data into different regions. Finally, the Riemann problem is resolved explicitly on a case-by-case basis.

Abstract We investigate the multiplicity of solutions for a quasilinear scalar field equation with a nonhomogeneous differential operator defined by S u ≔ − div ϕ u 2 + ∣ ∇ u ∣ 2 2 ∇ u + ϕ u 2 + ∣ ∇ u ∣ 2 2 u , Su:= -\hspace{0.1em}\text{div}\hspace{0.1em}\left\{\phi \left(\frac{{u}^{2}+{| \nabla u| }^{2}}{2}\right)\nabla u\right\}+\phi \left(\frac{{u}^{2}+{| \nabla u| }^{2}}{2}\right)u, where ϕ : [ 0 , + ∞ ) → R \phi :\left[0,+\infty )\to {\mathbb{R}} is a positive continuous function. This operator is introduced by Stuart [Two positive solutions of a quasilinear elliptic Dirichlet problem, Milan J. Math. 79 (2011), 327–341] and depends on not only ∇ u \nabla u but also u u . This particular quasilinear term generally appears in the study of nonlinear optics model, which describes the propagation of self-trapped beam in a cylindrical optical fiber made from a self-focusing dielectric material. When the reaction term is concave-convex nonlinearities, by using the Nehari manifold and doing a fine analysis associated on the fibering map, we obtain that the equation admits at least one positive energy solution and negative energy solution, which is also the ground state solution of the equation. We overcome two main difficulties which are caused by the nonhomogeneity of the differential operator S S : (i) the almost everywhere convergence of the gradient for the minimizing sequence { u n } \left\{{u}_{n}\right\} ; (ii) seeking the reasonable restrictions about S S .

Abstract The aim of this article is to study regularity properties of a local minimizer of a double phase functional of type ℱ ( u ) ≔ ∫ Ω ( ∣ D u ∣ p ( x ) + a ( x ) ∣ D u ∣ p ( x ) log ( e + ∣ D u ∣ ) ) d x , {\mathcal{ {\mathcal F} }}\left(u):= \mathop{\int }\limits_{\Omega }({| Du| }^{p\left(x)}+a\left(x){| Du| }^{p\left(x)}\log \left(e+| Du| )){\rm{d}}x, being p ( x ) , a ( x ) p\left(x),a\left(x) log-continuous functions with p ( x ) &gt; 1 p\left(x)\gt 1 , a ≥ 0 a\ge 0 . Double phase functionals ∫ ( ∣ D u ∣ p + a ( x ) ∣ D u ∣ q ) d x \int ({| Du| }^{p}+a\left(x){| Du| }^{q}){\rm{d}}x , with constant exponents p p and q q ( q ≥ p ≥ 1 ) \left(q\ge p\ge 1) , appeared in the papers by Zhikov, and C 1 , α {C}^{1,\alpha } -regularity of their minimizers was given by Colombo and Mingione. Later, by Baroni, Colombo, and Mingione, the above type functionals with logarithm but with constant exponent, regularity properties were given. They obtained sharp regularity results for minimizers of such functionals. In this article, we treat the case that the exponents are functions of p ( x ) p\left(x) and partly generalize their regularity results.

Abstract Using the Ekeland variational principle and the mountain pass lemma we prove some properties about potential homeomorphisms between a real Banach space and its dual. In particular, we show that a locally monotone homeomorphism is necessarily strictly monotone if it is potential.

Abstract In this article, we consider the following elliptic system of Hamiltonian-type on a bounded domain: − Δ u = K 1 ( ∣ y ∣ ) ∣ v ∣ p − 1 v , in B 1 ( 0 ) , − Δ v = K 2 ( ∣ y ∣ ) ∣ u ∣ q − 1 u , in B 1 ( 0 ) , u = v = 0 on ∂ B 1 ( 0 ) , \left\{\begin{array}{ll}-\Delta u={K}_{1}\left(| y| ){| v| }^{p-1}v,\hspace{1.0em}&amp; \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{B}_{1}\left(0),\\ -\Delta v={K}_{2}\left(| y| ){| u| }^{q-1}u,\hspace{1.0em}&amp; \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{B}_{1}\left(0),\\ u=v=0\hspace{1.0em}&amp; \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial {B}_{1}\left(0),\end{array}\right. where K 1 ( r ) {K}_{1}\left(r) and K 2 ( r ) {K}_{2}\left(r) are positive bounded functions defined in [ 0 , 1 ] \left[0,1] , B 1 ( 0 ) {B}_{1}\left(0) is the unit ball in R N {{\mathbb{R}}}^{N} , and ( p , q ) \left(p,q) is a pair of positive numbers lying on the critical hyperbola 1 p + 1 + 1 q + 1 = N − 2 N . \frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}. Under some suitable further assumptions on the functions K 1 ( r ) {K}_{1}\left(r) and K 2 ( r ) {K}_{2}\left(r) , we prove the existence of infinitely many nonradial positive solutions whose energy can be made arbitrarily large. Our proof is based on the reduction method. The most ingredients of the article are using the Green representation and estimating the Green function and its regular part very carefully. For this purpose, some more extra ideas and techniques are needed. We believe that our method and techniques can be applied to other related problems.

Abstract In this article, we consider the Cauchy problem for a semi-linear wave equation with time-dependent damping and memory nonlinearity. Conditions for global existence are presented in the energy space H 1 ( R n ) × L 2 ( R n ) {H}^{1}\left({{\mathbb{R}}}^{n})\times {L}^{2}\left({{\mathbb{R}}}^{n}) , n ≥ 1 n\ge 1 .

Abstract We study general problems modeling electrostatic microelectromechanical systems devices (Pλ ) φ ( r , − u ′ ( r ) ) = λ ∫ 0 r f ( s ) g ( u ( s ) ) d s , r ∈ ( 0 , 1 ) , 0 &lt; u ( r ) &lt; 1 , r ∈ ( 0 , 1 ) , u ( 1 ) = 0 , \left\{\begin{array}{ll}\varphi (r,-u^{\prime} \left(r))=\lambda \underset{0}{\overset{r}{\displaystyle \int }}\frac{f\left(s)}{g\left(u\left(s))}{\rm{d}}s,\hspace{1.0em}&amp; r\in \left(0,1),\\ 0\lt u\left(r)\lt 1,\hspace{1.0em}&amp; r\in \left(0,1),\\ u\left(1)=0,\hspace{1.0em}\end{array}\right. where φ \varphi , g g , and f f are some functions on [ 0 , 1 ] \left[0,1] and λ &gt; 0 \lambda \gt 0 is a parameter. We obtain results on the existence and regularity of a touchdown solution to ( P λ {P}_{\lambda } ) and find upper and lower bounds on the respective pull-in voltage. In the particular case, when φ ( r , v ) = r α ∣ v ∣ β v \varphi \left(r,v)={r}^{\alpha }{| v| }^{\beta }v , i.e., when the associated differential equation involves the operator r − γ ( r α ∣ u ′ ∣ β u ′ ) ′ {r}^{-\gamma }\left({r}^{\alpha }{| u^{\prime} | }^{\beta }u^{\prime} )^{\prime} , we obtain an exact asymptotic behavior of the touchdown solution in a neighborhood of the origin.

Abstract In this article, we show continuity of commutators of Calderón-Zygmund operators [ b , T ] \left[b,T] with BMO functions in generalized Orlicz-Morrey spaces M Φ , φ ( R n ) {M}^{\Phi ,\varphi }\left({{\mathbb{R}}}^{n}) . We give necessary and sufficient conditions for the boundedness of the genuine Calderón-Zygmund operators T T and for their commutators [ b , T ] \left[b,T] on generalized Orlicz-Morrey spaces, respectively.

Abstract In the present work, we establish a blow-up criterion for viscoelastic wave equations with nonlinear damping, logarithmic source, delay in the velocity, and acoustic boundary conditions. Due to the nonlinear damping term, we cannot apply the concavity method introduced by Levine. Thus, we use the energy method to show that the solution with negative initial energy blows up after finite time. Furthermore, we investigate the upper and lower bounds of the blow-up time.