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Register a new userMultiplicity of solutions for a class of quasilinear problems involving the $1$-Laplacian operator with critical growth
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The aim of this paper is to establish two results about multiplicity of solutions to problems involving the $1-$Laplacian operator, with nonlinearities with critical growth. To be more specific, we study the following problem $$ left{ begin{array}{l} - Delta_1 u +xi frac{u}{|u|} =lambda |u|^{q-2}u+|u|^{1^*-2}u, quadtext{in }Omega, u=0, quadtext{on } partialOmega. end{array} right. $$ where $Omega$ is a smooth bounded domain in $mathbb{R}^N$, $N geq 2$ and $xi in{0,1}$. Moreover, $lambda > 0$, $q in (1,1^*)$ and $1^*=frac{N}{N-1}$. The first main result establishes the existence of many rotationally non-equivalent and nonradial solutions by assuming that $xi=1$, $Omega = {x in mathbb{R}^N,:,r < |x| < r+1}$, $Ngeq 2$, $N ot = 3$ and $r > 0$. In the second one, $Omega$ is a smooth bounded domain, $xi=0$, and the multiplicity of solutions is proved through an abstract result which involves genus theory for functionals which are sum of a $C^1$ functional with a convex lower semicontinuous functional.
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In this paper we deal with the multiplicity of positive solutions to the fractional Laplacian equation begin{equation*} (-Delta)^{frac{alpha}{2}} u=lambda f(x)|u|^{q-2}u+|u|^{2^{*}_{alpha}-2}u, quadtext{in},,Omega, u=0,text{on},,partialOmega, end{equation*} where $Omegasubset mathbb{R}^{N}(Ngeq 2)$ is a bounded domain with smooth boundary, $0<alpha<2$, $(-Delta)^{frac{alpha}{2}}$ stands for the fractional Laplacian operator, $fin C(Omegatimesmathbb{R},mathbb{R})$ may be sign changing and $lambda$ is a positive parameter. We will prove that there exists $lambda_{*}>0$ such that the problem has at least two positive solutions for each $lambdain (0,,,lambda_{*})$. In addition, the concentration behavior of the solutions are investigated.
We investigate the existence of infinitely many radially symmetric solutions to the following problem $$(-Delta_p)^s u=g(u) textrm{ in } mathbb{R}^N, uin W^{s,p}(mathbb{R}^N),$$ where $sin (0,1)$, $2 leq p < infty$, $sp leq N $, $2 leq N in mathbb{N}$ and $(-Delta_p)^s$ is the fractional $p$-Laplacian operator. We treat both of cases $sp=N$ and $sp<N.$ The nonlinearity $g$ is a function of Berestycki-Lions type with critical exponential growth if $sp=N$ and critical polynomial growth if $sp<N$. We also prove the existence of a ground state solution for the same problem.
It is establish regularity results for weak solutions of quasilinear elliptic problems driven by the well known $Phi$-Laplacian operator given by begin{equation*} left{ begin{array}{cl} displaystyle-Delta_Phi u= g(x,u), & mbox{in}~Omega, u=0, & mbox{on}~partial Omega, end{array} right. end{equation*} where $Delta_{Phi}u :=mbox{div}(phi(| abla u|) abla u)$ and $Omegasubsetmathbb{R}^{N}, N geq 2,$ is a bounded domain with smooth boundary $partialOmega$. Our work concerns on nonlinearities $g$ which can be homogeneous or non-homogeneous. For the homogeneous case we consider an existence result together with a regularity result proving that any weak solution remains bounded. Furthermore, for the non-homogeneous case, the nonlinear term $g$ can be subcritical or critical proving also that any weak solution is bounded. The proofs are based on Mosers iteration in Orclicz and Orlicz-Sobolev spaces.
The authors of this paper study singular phenomena(vanishing and blowing-up in finite time) of solutions to the homogeneous $hbox{Dirichlet}$ boundary value problem of nonlinear diffusion equations involving $p(x)$-hbox{Laplacian} operator and a nonlinear source. The authors discuss how the value of the variable exponent $p(x)$ and initial energy(data) affect the properties of solutions. At the same time, we obtain the critical extinction and blow-up exponents of solutions.
The main goal of this paper is the study of two kinds of nonlinear problems depending on parameters in unbounded domains. Using a nonstandard variational approach, we first prove the existence of bounded solutions for nonlinear eigenvalue problems involving the fractional Laplace operator and nonlinearities that have subcritical growth. In the second part, based on a variational principle of Ricceri [16], we study a fractional nonlinear problem with two parameters and prove the existence of multiple solutions.
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