Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userInfinitely many solutions of a class of elliptic equations with variable exponent
148
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
This paper is concerned with the $p(x)$-Laplacian equation of the form begin{equation}label{eq0.1} left{begin{array}{ll} -Delta_{p(x)} u=Q(x)|u|^{r(x)-2}u, &mbox{in} Omega, u=0, &mbox{on} partial Omega, end{array}right. end{equation} where $OmegasubsetR^N$ is a smooth bounded domain, $1<p^-=min_{xinoverline{Omega}}p(x)leq p(x)leqmax_{xinoverline{Omega}}p(x)=p^+<N$, $1leq r(x)<p^{*}(x)=frac{Np(x)}{N-p(x)}$, $r^-=min_{xin overline{Omega}}r(x)<p^-$, $r^+=max_{xinoverline{Omega}}r(x)>p^+$ and $Q: overline{Omega}toR$ is a nonnegative continuous function. We prove that eqref{eq0.1} has infinitely many small solutions and infinitely many large solutions by using the Clarks theorem and the symmetric mountain pass lemma.
rate research
Read More
We prove the existence of infinitely many nonnegative solutions to the following nonlocal elliptic partial differential equation involving singularities begin{align} (-Delta)_{p(cdot)}^{s} u&=frac{lambda}{|u|^{gamma(x)-1}u}+f(x,u)~text{in}~Omega, onumber u&=0~text{in}~mathbb{R}^NsetminusOmega, onumber end{align} where $Omegasubsetmathbb{R}^N,, Ngeq2$ is a smooth, bounded domain, $lambda>0$, $sin (0,1)$, $gamma(x)in(0,1)$ for all $xinbar{Omega}$, $N>sp(x,y)$ for all $(x,y)inbar{Omega}timesbar{Omega}$ and $(-Delta)_{p(cdot)}^{s}$ is the fractional $p(cdot)$-Laplacian operator with variable exponent. The nonlinear function $f$ satisfies certain growth conditions. Moreover, we establish a uniform $L^{infty}(bar{Omega})$ estimate of the solution(s) by the Moser iteration technique.
In this paper, we consider the existence and asymptotic properties of solutions to the following Kirchhoff equation begin{equation}label{1} onumber - Bigl(a+bint_{{R^3}} {{{left| { abla u} right|}^2}}Bigl) Delta u =lambda u+ {| u |^{p - 2}}u+mu {| u |^{q - 2}}u text { in } mathbb{R}^{3} end{equation} under the normalized constraint $int_{{mathbb{R}^3}} {{u}^2}=c^2$, where $a!>!0$, $b!>!0$, $c!>!0$, $2!<!q!<!frac{14}{3}!<! p!leq!6$ or $frac{14}{3}!<!q!<! p!leq! 6$, $mu!>!0$ and $lambda!in!R$ appears as a Lagrange multiplier. In both cases for the range of $p$ and $q$, the Sobolev critical exponent $p!=!6$ is involved and the corresponding energy functional is unbounded from below on $S_c=Big{ u in H^{1}({mathbb{R}^3}): int_{{mathbb{R}^3}} {{u}^2}=c^2 Big}$. If $2!<!q!<!frac{10}{3}$ and $frac{14}{3}!<! p!<!6$, we obtain a multiplicity result to the equation. If $2!<!q!<!frac{10}{3}!<! p!=!6$ or $frac{14}{3}!<!q!<! p!leq! 6$, we get a ground state solution to the equation. Furthermore, we derive several asymptotic results on the obtained normalized solutions. Our results extend the results of N. Soave (J. Differential Equations 2020 $&$ J. Funct. Anal. 2020), which studied the nonlinear Schr{o}dinger equations with combined nonlinearities, to the Kirchhoff equations. To deal with the special difficulties created by the nonlocal term $({int_{{R^3}} {left| { abla u} right|} ^2}) Delta u$ appearing in Kirchhoff type equations, we develop a perturbed Pohozaev constraint approach and we find a way to get a clear picture of the profile of the fiber map via careful analysis. In the meantime, we need some subtle energy estimates under the $L^2$-constraint to recover compactness in the Sobolev critical case.
We study boundary blow-up solutions of semilinear elliptic equations $Lu=u_+^p$ with $p>1$, or $Lu=e^{au}$ with $a>0$, where $L$ is a second order elliptic operator with measurable coefficients. Several uniqueness theorems and an existence theorem are obtained.
For a general class of autonomous quasi-linear elliptic equations on R^n we prove the existence of a least energy solution and show that all least energy solutions do not change sign and are radially symmetric up to a translation in R^n.
This paper is devoted to the study of periodic (in time) solutions to an one-dimensional semilinear wave equation with $x$-dependent coefficients under various homogeneous boundary conditions. Such a model arises from the forced vibrations of a nonhomogeneous string and propagation of seismic waves in nonisotropic media. By combining variational methods with an approximation argument, we prove that there exist infinitely many periodic solutions whenever the period is a rational multiple of the length of the spatial interval. The proof is essentially based on the spectral properties of the wave operator with $x$-dependent coefficients.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
