We study a Dirichlet type problem for an equation involving the fractional Laplacian and a reaction term subject to either subcritical or critical growth conditions, depending on a positive parameter. Applying a critical point result of Bonanno, we prove existence of one or two positive solutions as soon as the parameter lies under a (explicitly determined) threshold. As an application, we find two positive solutions for a fractional Brezis-Nirenberg problem.
We prove global well-posedness of strong solutions and existence of the global attractor for the 2D Boussinesq system in a periodic channel with fractional Laplacian in subcritical case. The analysis reveals a relation between the Laplacian exponent and the regularity of the spaces of velocity and temperature.
We consider the following boundary value problem -Δu= g(x,u) + f(x,u) x\in Ω u=0 x\in \partial Ω where $g(x,-ξ)=-g(x,ξ)$ and $g$ has subcritical exponential growth in $\mathbb{R} ^2$. Using the method developed by Bolle, we prove that this problem has infinitely many solutions under suitable conditions on the growth of $g(u)$ and $f(u)$.
In this paper we deal with a Dirichlet problem for an elliptic equation involving the $1$-Laplacian operator and a source term. We prove that, when the growth of the source is subcritical, there exist two bounded nontrivial solutions to our problem. Moreover, a Pohozaev type identity is proved, which holds even when the growth is supercritical. We also show explicit examples of our results.
In the present work we obtain the existence and multiplicity of nontrivial solutions for the Choquard logarithmic equation $(-Δ)_{p}^{s}u + |u|^{p-2}u + (\ln|\cdot|\ast |u|^{p})|u|^{p-2}u = f(u) \textrm{ \ in \ } \mathbb{R}^N $ , where $ N=sp $, $ s\in (0, 1) $, $ p>2 $, $ a>0 $, $ λ>0 $ and $f: \mathbb{R}\rightarrow \mathbb{R} $ a continuous nonlinearity with exponential critical and subcritical growth. We guarantee the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution under critical and subcritical growth. Morever, when $ f $ has subcritical…
In this paper, we consider the existence of nodal solutions with two bubbles to the slightly subcritical problem with the fractional Laplacian \begin{equation*} \left\{\aligned &(-Δ)^su=|u|^{p-1-\varepsilon}u\ \ \mbox{in}\ Ω&u=0\ \mbox{on}\ \partialΩ, \endaligned \right. \end{equation*} where $Ω$ is a smooth bounded domain in $\mathbb R^N$, $N>2s$, $0<s<1$, $ p=\frac{N+2s}{N-2s}$ and $\varepsilon>0$ is a small parameter, which can be seen as a nonlocal analog of the results of Bartsch, Micheletti and Pistoia (2006) \cite{Bartsch1}.
We consider an elliptic problem with nonlinear boundary condition involving nonlinearity with superlinear and subcritical growth at infinity and a bifurcation parameter as a factor. We use re-scaling method, degree theory and continuation theorem to prove that there exists a connected branch of positive solutions bifurcating from infinity when the parameter goes to zero. Moreover, if the nonlinearity satisfies additional conditions near zero, we establish a global bifurcation result, and discuss the number of positive solution(s) with respect to the parameter using bifurcation theory and degre…
Let $Ω$ be a bounded domain in $\mathbb{R}^N$. In this paper, we consider the following nonlinear elliptic equation of $N$-Laplacian type: $-Δ_{N}u=f(x,u)$ where $u\in W_{0}^{1,2}\{0}$ when $f$ is of subcritical or critical exponential growth. This nonlinearity is motivated by the Moser-Trudinger inequality. In fact, we will prove the existence of a nontrivial nonnegative solution to the above equation without the Ambrosetti-Rabinowitz $(AR)$ condition. Earlier works in the literature on the existence of nontrivial solutions to $N-$Laplacian in $\mathbb{R}^{N}$ when the nonlinear term $f$ has …
In this paper, we consider the following fractional Laplacian system with one critical exponent and one subcritical exponent \begin{equation*} \begin{cases} (-Δ)^{s}u+μu=|u|^{p-1}u+λv & x\in \ \mathbb{R}^{N}, (-Δ)^{s}v+νv = |v|^{2^{\ast}-2}v+λu& x\in \ \mathbb{R}^{N},\\ \end{cases} \end{equation*} where $(-Δ)^{s}$ is the fractional Laplacian, $0<s<1,\ N>2s, \ λ<\sqrt{μν},\ 1<p<2^{\ast}-1~ and~\ 2^{\ast}=\frac{2N}{N-2s}$~ is the Sobolev critical exponent. By using the Nehari\ manifold, we show that there exists a $μ_{0}\in(0,1)$, such that when $0<μ\leqμ_{0}$, the system has a positive ground s…
In this paper, we study the multiplicity of positive solutions for the p-Laplacian systems with sign-changing weight functions. Using the decomposition of the Nehari manifold, we prove that an elliptic system has at least two positive solutions.
We study qualitative properties of solutions to the fractional Lane-Emden-Fowler equations with slightly subcritical exponents where the associated fractional Laplacian is defined in terms of either the spectra of Dirichlet Laplacian or the integral representation. As a consequence, we classify the asymptotic behavior of all finite energy solutions. Our method also provides a simple and unified approach to deal with the classical (local) Lane-Emden-Fowler equation for any dimension greater than 2.
We prove new multiplicity results for the Brezis-Nirenberg problem for the $p$-Laplacian. Our proofs are based on a new abstract critical point theorem involving the ${\mathbb Z}_2$-cohomological index that requires less compactness than the (PS) condition.
We obtain fundamental imbeddings for the fractional Sobolev space with variable exponent that is a generalization of well-known fractional Sobolev spaces. As an application, we obtain a-priori bounds and multiplicity of solutions to some nonlinear elliptic problems involving the fractional $p(\cdot)$-Laplacian.
We prove a bifurcation and multiplicity result that is independent of the dimension N for a critical p-Laplacian problem that is the analog of the Brezis-Nirenberg problem for the quasilinear case. This extends a result in the literature for the semilinear case.
We present new $L^\infty$ a priori estimates for weak solutions of a wide class of subcritical $p$-laplacian equations in bounded domains. No hypotheses on the sign of the solutions, neither of the non-linearities are required. This method is based in elliptic regularity for the $p$-laplacian combined either with Gagliardo-Nirenberg or Caffarelli-Kohn-Nirenberg interpolation inequalities. Let us consider a quasilinear boundary value problem $ -Δ_p u= f(x,u),$ in $Ω,$ with Dirichlet boundary conditions, where $Ω\subset \mathbb{R}^N $, with $p<N,$ is a bounded smooth domain strictly convex, and …
We consider a class of parametric Schrödinger equations driven by the fractional $p$-Laplacian operator and involving continuous positive potentials and nonlinearities with subcritical or critical growth. By using variational methods and Ljusternik-Schnirelmann theory, we study the existence, multiplicity and concentration of positive solutions for small values of the parameter.
We prove the existence of $N$ distinct pairs of nontrivial solutions for critical $p$-Laplacian problems in ${\mathbb R}^N$, as well as in bounded domains. To overcome the difficulties arising from the lack of compactness, we use a recent global compactness result of Mercuri and Willem.
In this paper, first we study existence results for a linearly perturbed elliptic problem driven by the fractional Laplacian. Then, we show a multiplicity result when the perturbation parameter is close to the eigenvalues. This latter result is obtained by exploiting the topological structure of the sublevels of the associated functional, which permits to apply a critical point theorem of mixed nature due to Marino and Saccon.
In this paper we investivate bifurcation results for a class of problem in a smooth bounded domain involving the fractional p-Laplacian operator and with a nonlinearity that reaches the critical growth with respect to the fractional Sobolev embedding.
In this paper we establish the multiplicity of nontrivial weak solutions for the problem $(-Δ)^α u +u= h(u)$ in $Ω_λ$,\ $u=0$ on $\partialΩ_λ$, where $Ω_λ=λΩ$, $Ω$ is a smooth and bounded domain in $\mathbb{R}^N, N>2α$, $λ$ is a positive parameter, $α\in (0,1)$, $(-Δ)^α$ is the fractional Laplacian and the nonlinear term $h(u)$ has a subcritical growth. We use minimax methods, the Ljusternick-Schnirelmann and Morse theories to get multiplicity result depending on the topology of $Ω$.
We study the incompressible Navier-Stokes equations with a fractional Laplacian and prove the existence of discontinuous Leray-Hopf solutions in the largest critical space with arbitrarily small initial data.
In this article, we study the existence of positive solutions to elliptic equation (E1) $$(-Δ)^αu=g(u)+σν\quad{\rm in}\quad Ω,$$ subject to the condition (E2) $$u=\varrhoμ\quad {\rm on}\quad \partialΩ \ {\rm if}\ α=1\qquad {\rm or\ \ in}\ \ Ω^c \ \ {\rm if}\ α\in(0,1),$$ where $σ,\varrho\ge0$, $Ω$ is an open bounded $C^2$ domain in $\mathbb{R}^N$, $(-Δ)^α$ denotes the fractional Laplacian with $α\in(0,1)$ or Laplacian operator if $α=1$, $ν,μ$ are suitable Radon measures and $g:\mathbb{R}_+\mapsto\mathbb{R}_+$ is a continuous function. We introduce an approach to obtain weak solutions for probl…
The paper is concerned with the existence of positive weak solutions for a new class of $\left( p,q\right) $-Laplacian elliptic systems in a bounded domain by means of the method of sub-super solutions. Particularly, we do not need any sign conditions for $γ\left( 0\right), g\left( 0\right), f\left( 0\right) $ and $h\left(0\right) $. Moreover, a multiplicity result is obtained when $γ\left(0\right)=g\left( 0\right)=f\left( 0\right)=h\left( 0\right)=0.$ Finally, we give some examples to verify our main results.
In this paper, we deal with the existence and multiplicity of solutions to the nonuniformly elliptic equation of the N-Lapalcian type with a potential and a nonlinear term of critical exponential growth and satisfying the Ambrosetti-Rabinowitz condition. In spite of a possible failure of the Palais-Smale compactness condition, in this article we apply minimax method to obtain the weak solution to such an equation. In particular, in the case of $N-$Laplacian, using the minimization and the Ekeland variational principle, we obtain multiplicity of weak solutions. Finally, we will prove the above …
In this paper we deal with the multiplicity of positive solutions to the fractional Laplacian equation \begin{equation*} (-Δ)^{\fracα{2}} u=λf(x)|u|^{q-2}u+|u|^{2^{*}_α-2}u, \quad\text{in}\,\,Ω, u=0,\text{on}\,\,\partialΩ, \end{equation*} where $Ω\subset \mathbb{R}^{N}(N\geq 2)$ is a bounded domain with smooth boundary, $0<α<2$, $(-Δ)^{\fracα{2}}$ stands for the fractional Laplacian operator, $f\in C(Ω\times\mathbb{R},\mathbb{R})$ may be sign changing and $λ$ is a positive parameter. We will prove that there exists $λ_{*}>0$ such that the problem has at least two positive solutions for each $λ…
In this paper we develop a new theory for the existence, localization and multiplicity of positive solutions for a class of non-variational,quasilinear, elliptic systems. In order to do this, we provide a fairly general abstract framework for the existence of fixed points of nonlinear operators acting on cones that satisfy an inequality of Harnack type. Our methodology relies on fixed point index theory. We also provide a non-existence result and an example to illustrate the theory.
In this short note, we consider the elliptic problem $$ λϕ+ Δϕ= η|ϕ|^σϕ,\quad ϕ\big|_{\partial Ω}=0,\quad λ, η\in \mathbb{C}, $$ on a smooth domain $Ω\subset \mathbb{R}^N$, $N\ge 1$. The presence of complex coefficients, motivated by the study of complex Ginzburg-Landau equations, breaks down the variational structure of the equation. We study the existence of nontrivial solutions as bifurcations from the trivial solution. More precisely, we characterize the bifurcation branches starting from eigenvalues of the Dirichlet-Laplacian of arbitrary multiplicity. This allows us to discuss the nature…
In this article, we prove the finite dimensionality of the global attractor and estimate the numbers of the determining modes for the 2D Boussinesq system in a periodic channel with fractional Laplacian in subcritical case.
In this work we consider the following class of fractional $p\&q$ Laplacian problems \begin{equation*} (-Δ)_{p}^{s}u+ (-Δ)_{q}^{s}u + V(\varepsilon x) (|u|^{p-2}u + |u|^{q-2}u)= f(u) \mbox{ in } \mathbb{R}^{N}, \end{equation*} where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $1< p<q<\frac{N}{s}$, $(-Δ)^{s}_{t}$, with $t\in \{p,q\}$, is the fractional $t$-Laplacian operator, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a continuous potential and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a $\mathcal{C}^{1}$-function with subcritical growth. Applying minimax theorems and the Ljusternik-Schnirelm…
In the present work we investigate the existence and multiplicity of nontrivial solutions for the Choquard Logarithmic equation $(-Δ)^{\frac{1}{2}} u + au + λ(\ln|\cdot|\ast |u|^{2})u = f(u) \textrm{ in } \mathbb{R}$, for $ a>0 $, $ λ>0 $ and a nonlinearity $f$ with exponential critical growth. We prove the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution under exponential critical and subcritical growth. Morever, when $ f $ has subcritical growth we guarantee the existence of infinitely many solutions, via genus theory.
It is well known that a single nonlinear fractional Schrödinger equation with a potential $V(x)$ and a small parameter $\varepsilon $ may have a positive solution that is concentrated at the nondegenerate minimum point of $V(x)$. In this paper, we can find two different positive solutions for two weakly coupled fractional Schrödinger systems with a small parameter $\varepsilon $ and two potentials $V_{1}(x)$ and $V_{2}(x)$ having the same minimum point are concentrated at the same point minimum point of $V_{1}(x)$ and $V_{2}\left(x\right) $. In fact that by using the energy estimates, Nehari m…
We investigate the existence of infinitely many radially symmetric solutions to the following problem $$(-Δ_p)^s u=g(u) \ \ \textrm{ in } \ \ \mathbb{R}^N, \ \ u\in W^{s,p}(\mathbb{R}^N),$$ where $s\in (0,1)$, $2 \leq p < \infty$, $sp \leq N $, $2 \leq N \in \mathbb{N}$ and $(-Δ_p)^s$ 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.
The aim of this paper is to obtain the existence of solutions for the following fractional p-Laplacian Dirichlet problem with mixed derivatives \begin{eqnarray*} &{_{t}}D_{T}^α\left(|_{0}D_{t}^αu(t)|^{p-2}{_{0}}D_{t}^αu(t)\right) = f(t, u(t)), \;t\in [0,T],\\ &u(0) = u(T) = 0, \end{eqnarray*} where $0 < α<1$, $1<p<\infty$ and $f:[0,T]\times \mathbb{R} \to \mathbb{R}$ is a continuous function. We obtain the existence of nontrivial solutions by using the direct method in variational methods and the genus in the critical point theory. Furthermore, if $0< α< \frac{1}{p}$ we obtain an almost every …
We consider the boundary value problem $-Δ_p u = λc(x) |u|^{p-2}u + μ(x) |\grad u|^p + h(x)$, $u \in W^{1,p}_0(Ω) \cap L^{\infty}(Ω)$, where $Ω\subset \mathbb R^N$, $N \geq 2$, is a bounded domain with smooth boundary. We assume $c$, $h \in L^q(Ω)$ for some $q > \max\{N/p,1\}$ with $c \gneqq 0$ and $μ\in L^{\infty}(Ω)$. We prove existence and uniqueness results in the coercive case $ λ\leq 0$ and existence and multiplicity results in the non-coercive case $ λ>0$. Also, considering stronger assumptions on the coefficients, we clarify the structure of the set of solutions in the non-coercive cas…
In this paper, we consider the semilinear equation involving the fractional Laplacian in the Euclidian space $\mathbb{R}^n$: \begin{equation} (-Δ)^{α/2} u(x) = f(x_n) \,u^p(x), \quad x \in \mathbb{R}^n \label{n26} \end{equation} in the subcritical case with $1<p<\frac{n+α}{n-α}$. Instead of carrying out direct investigations on pseudo-differential equation (\ref{n26}), we first seek its equivalent form in an integral equation as below: \begin{equation} u(x)=\int_{\mathbb{R}^n}G_{\infty}(x,y)\,f(y_n)\, u^{p}(y)\,dy, \label{n27} \end{equation} where $ G_{\infty}(x,y)$ is the Green's function ass…
In this paper, we deal with a singular quasilinear critical elliptic equation of Lichnerowicz type involving the p-Laplacian operator. With the help of the subcritical approach from variational method, we obtain the non-existence, existence, and multiplicity results under some given assumptions.
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 $$ \left\{ \begin{array}{l} - Δ_1 u +ξ\frac{u}{|u|} =λ|u|^{q-2}u+|u|^{1^*-2}u, \quad\text{in }Ω, u=0, \quad\text{on } \partialΩ. \end{array} \right. $$ where $Ω$ is a smooth bounded domain in $\mathbb{R}^N$, $N \geq 2$ and $ξ\in\{0,1\}$. Moreover, $λ> 0$, $q \in (1,1^*)$ and $1^*=\frac{N}{N-1}$. The first main result establishes the existence of many rotationally non-equi…
In this paper, we study the existence of three solutions for a Kirchhoff equation involving the nonlocal fractional p-Laplacian considering Sobolev and Hardy nonlinearities at subcritical and critical growths. The proof is based on Mountain Pass Theorem, Ekeland's variational principle and constrained minimization in Nehari sets.
In this paper, we consider a p-Laplacian equation in {R}^{N}with sign-changing potential and subcritical p-superlinear nonlinearity. By using the cohomological linking method for cones developed by Degiovanni and Lancelotti in 2007, an existence result is obtained. We also give a result on the existence of periodic solutions for one-dimensional $p$-Laplacian equations which can be proved by the same method.
In this paper we study the multiplicity and concentration of positive solutions for the following $(p, q)$-Laplacian problem: \begin{equation*} \left\{ \begin{array}{ll} -Δ_{p} u -Δ_{q} u +V(\varepsilon x) \left(|u|^{p-2}u + |u|^{q-2}u\right) = f(u) &\mbox{ in } \mathbb{R}^{N}, \\ u\in W^{1, p}(\mathbb{R}^{N})\cap W^{1, q}(\mathbb{R}^{N}), \quad u>0 \mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $1< p<q<N$, $Δ_{r}u=\mbox{div}(|\nabla u|^{r-2}\nabla u)$, with $r\in \{p, q\}$, is the $r$-Laplacian operator, $V:\mathbb{R}^{N}\rightarrow…
In this paper, we establish the results on the existence, nonexistence and multiplicity of positive solutions to singular boundary value problems involving $φ$-Laplacian. Our approach is based on the fixed point index theory. The interesting point is that a result for the existence of three positive solutions is given.
Let $1<p<+\infty$ and let $Ω\subset\mathbb R^N$ be either a ball or an annulus. We continue the analysis started in [Boscaggin, Colasuonno, Noris, ESAIM Control Optim. Calc. Var. (2017)], concerning quasilinear Neumann problems of the type \[ -Δ_p u = f(u), \quad u>0 \mbox{ in } Ω, \quad \partial_νu = 0 \mbox{ on } \partialΩ. \] We suppose that $f(0)=f(1)=0$ and that $f$ is negative between the two zeros and positive after. In case $Ω$ is a ball, we also require that $f$ grows less than the Sobolev-critical power at infinity. We prove a priori bounds of radial solutions, focusing in particular…
In this paper we study the number of the boundary single peak solutions of the problem {align*} {cases} -\varepsilon^2 Δu + u = u^p, &\text{in}Ωu > 0, &\text{in}Ω\frac{\partial u}{\partial ν} = 0,& \text{on}\partial Ω{cases} {align*} for $\varepsilon$ small and $p$ subcritical. Under some suitable assumptions on the shape of the boundary near a critical point of the mean curvature, we are able to prove exact multiplicity results. Note that the degeneracy of the critical point is allowed.
We consider Dirichlet elliptic equations driven by the sum of a $p$-Laplacian $(2<p)$ and a Laplacian. The conditions on the reaction term imply that the problem is resonant at both $\pm\infty$ and at zero. We prove an existence theorem (producing one nontrivial smooth solution) and a multiplicity theorem (producing five nontrivial smooth solutions, four of constant sign and the fifth nodal; the solutions are ordered). Our approach uses variational methods and critical groups.
In the present paper we deal with a quasilinear problem involving a singular term and a parametric superlinear perturbation. We are interested in the existence, nonexistence and multiplicity of positive solutions as the parameter $λ>0$ varies. In our first result, the superlinear perturbation has an arbitrary growth and we obtain the existence of a solution for the problem by using the sub-supersolution method. For the second result, the superlinear perturbation has subcritical growth and we employ the Mountain Pass Theorem to show the existence of a second solution.
We prove the stability of the Einstein-scalar field Lichnerowicz equation under subcritical perturbations of the critical nonlinearity in dimensions 3, 4 and 5. As a consequence, we obtain the existence of a second solution to the equation in several cases. In particular, in the positive case, including the CMC positive cosmological constant case, we show that each time a solution exists, the equation produces a second solution with the exception of one critical value for which the solution is unique.
In this article we are interested in the following fractional $p$-Laplacian equation in $\mathbb{R}^n$ \begin{eqnarray*} &(-Δ)_{p}^αu + V(x)u^{p-2}u = f(x,u) \mbox{ in } \mathbb{R}^{n}, \end{eqnarray*} where $p\geq 2$, $0< s < 1$, $n\geq 2$ and subcritical p-superlinear nonlinearity. By using mountain pass theorem with Cerami condition we prove the existence of nontrivial solution. Furthermore, we show that this solution is radially simmetry.
we study on compact Riemannian manifolds with boundary, the problems of existence and multiplicity of solutions to a Neumann problem involving the p-Laplacian operator and critical Sobolev exponents.
We consider the subcritical SQG equation in its natural scale invariant Sobolev space and prove the existence of a global attractor of optimal regularity. The proof is based on a new energy estimate in Sobolev spaces to bootstrap the regularity to the optimal level, derived by means of nonlinear lower bounds on the fractional laplacian. This estimate appears to be new in the literature, and allows a sharp use of the subcritical nature of the $L^\infty$ bounds for this problem. As a byproduct, we obtain attractors for weak solutions as well. Moreover, we study the critical limit of the attracto…
