On a Riemannian compact manifold, we give existence and multiplicity results for solutions of elliptic PDE by introducing isometry invariances. When the groups we used have finite orbits, we get multiplicity results for equations with the classical critical Sobolev exponent, for instance the Yamabe equation. When there is no finite orbits, the multiplicity is obtained for equations with overcritical exponents.
We establish the multiplicity of positive solutions to a quasilinear Neumann problem in expanding balls and hemispheres with critical exponent in the boundary condition.
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 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 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.
In this paper we study the existence of at least two positive weak solutions for an inhomogeneous fourth order equation with Navier boundary data involving nonlinearities of critical growth with a bifurcation parameter $λ$ in $\mathbb{R}^{2m}$. We establish here the lower and upper bound for $λ$ which determine multiplicity and non-existence respectively.
In this paper we prove the existence of infinitely many nontrivial solutions for the class of $(p,\, q)$ fractional elliptic equations involving concave-critical nonlinearities in bounded domains in $\mathbb{R}^N$. Further, when the nonlinearity is of convex-critical type, we establish the multiplicity of nonnegative solutions using variational methods. In particular, we show the existence of at least $cat_Ω(Ω)$ nonnegative solutions.
In this work, we study the existence and multiplicity of solutions for a class of problems involving the $ϕ$-Laplacian operator in a bounded domain, where the nonlinearity has a critical growth. The main tool used is the variational method combined with the genus theory for even functionals.
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, we establish the existence and multiplicity of multi-bump nodal solutions for the following class of problems $$ -Δu+(λV(x)+1)u=f(u),~~\mbox{in}~~\mathbb{R}^2, $$ where $λ\in(0,\infty)$, $f$ is a continuous function with exponential critical growth and $V:\mathbb{R}^2\to \mathbb{R}$ is a continuous function verifying some hypotheses.
In this paper we study the fractional analogue of the Allen-Cahn energy in bounded domains, and we show that it admits a number of critical points which goes to infinity as the perturbation parameter tends to zero.
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.
This paper deals with multiplicity and bifurcation results for nonlinear problems driven by the fractional Laplace operator $(-Δ)^s$ and involving a critical Sobolev term. In particular, we consider $$\begin{cases} (-Δ)^su=γ|u|^{2^*-2}u+f(x,u) & \mbox{in } Ωu=0 & \mbox{in } \mathbb R^n\setminus Ω, \end{cases}$$ where $Ω\subset\mathbb R^n$ is an open bounded set with continuous boundary, $n>2s$ with $s\in(0,1)$, $γ$ is a positive real parameter, $2^*=2n/(n-2s)$ is the fractional critical Sobolev exponent and $f$ is a Carathéodory function satisfying different subcritical conditions.
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.
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 …
It is established some existence and multiplicity of solution results for a quasilinear elliptic problem driven by $Φ$-Laplacian operator. One of these solutions is built as a ground state solution. In order to prove our main results we apply the Nehari method combined with the concentration compactness theorem in an Orlicz-Sobolev framework. One of the difficulties in dealing with this kind of operator is the lost of homogeneity properties.
In this work, we prove the existence and multiplicity of positive solutions for the following class of quasilinear elliptic equations $$ \left \{ \begin{array}{lll} -ε^{N}Δ_{N} u + \left(1+μA(x) \right)\left| u\right|^{N-2}u= f(u)\,\,\,\, \mbox{ in } \, \RR^{N},\\ u > 0 \,\,\,\, \mbox{ in } \, \RR^{N}, \end{array} \right. $$ \\ where $Δ_{N}$ is the N-Laplacian operator, $N \geq 2$, $f$ is a function with exponential critical growth, $μ$ and $ε$ are positive parameters and $A$ is a nonnegative continuous function verifying some hypotheses. To obtain our results, we combine variational arguments…
In this paper we study the following nonlinear Schrödinger equation with magnetic field \[ \Big(\frac{\varepsilon}{i}\nabla-A(x)\Big)^{2}u+V(x)u=f(| u|^{2})u,\quad x\in\mathbb{R}^{2}, \] where $\varepsilon>0$ is a parameter, $V:\mathbb{R}^{2}\rightarrow \mathbb{R}$ and $A: \mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ are continuous potentials and $f:\mathbb{R}\rightarrow \mathbb{R}$ has exponential critical growth. Under a local assumption on the potential $V$, by variational methods, penalization technique, and Ljusternick-Schnirelmann theory, we prove multiplicity and concentration of solutions…
In order to obtain solutions to problem $$ {{array}{c} -Δu=\dfrac{A+h(x)} {|x|^2}u+k(x)u^{2^*-1}, x\in {\mathbb R}^N, u>0 \hbox{in}{\mathbb R}^N, {and}u\in {\mathcal D}^{1,2}({\mathbb R}^N), {array}. $$ $h$ and $k$ must be chosen taking into account not only the size of some norm but the shape. Moreover, if $h(x)\equiv 0$, to reach multiplicity of solution, some hypotheses about the local behaviour of $k$ close to the points of maximum are needed.
In this paper we study the multiplicity of normalized solutions to the following nonlinear Schrödinger equation with critical growth \begin{align*} \left\{ \begin{aligned} &-Δu=λu+μ|u|^{q-2}u+f(u), \quad \quad \hbox{in }\mathbb{R}^N,\\ &\int_{\mathbb{R}^{N}}|u|^{2}dx=a^{2}, \end{aligned} \right. \end{align*} where $a,μ>0$, $λ\in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier, $q \in (2,2+\frac{4}{N})$ and $f$ has an exponential critical growth when $N=2$, and $f(u)=|u|^{2^*-2}u$ when $N \geq 3$ and $2^{*}=\frac{2N}{N-2}$.
In this work we study an existence and multiplicity result for the following prescribed mean-curvature problem with critical growth $$ \left\{\begin{array}{rl} -\mbox{div}\biggl(\frac{\nabla u}{\sqrt{1+|\nabla u|^{2}}}\biggl) = λ|u|^{q-2}u+ |u|^{2^*-2}u & \mbox{in $Ω$} u = 0 & \mbox{on $\partial Ω$}, \end{array} \right. $$ where $Ω$ is a bounded smooth domain of $\mathbb{R}^{N}$, $N\geq 3$ and $1 < q<2$. In order to employ variational arguments, we consider an auxiliary problem which is proved to have infinitely many solutions by genus theory. A clever estimate in the gradient of the solutions…
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 study the multiplicity and concentration of the positive solutions to the following critical Kirchhoff type problem: \begin{equation*} -\left(\varepsilon^2 a+\varepsilon b\int_{\R^3}|\nabla u|^2\mathrm{d} x\right)Δu + V(x) u = f(u)+u^5\ \ {\rm in } \ \ \R^3, \end{equation*} where $\varepsilon$ is a small positive parameter, $a$, $b$ are positive constants, $V \in C(\mathbb{R}^3)$ is a positive potential, $f \in C^1(\R^+, \R)$ is a subcritical nonlinear term, $u^5$ is a pure critical nonlinearity. When $\varepsilon>0$ small, we establish the relationship between the number of …
In this paper we consider the functional whose critical points are solutions of the fractional CR Yamabe type equation on the sphere. We firstly study the behavior of the Palais-Smale sequences characterizing the bubbling phenomena and therefore we prove a multiplicity type result by showing the existence of infinitely many solutions to the related equation.
In this paper we prove an existence result of multiple positive solutions for the following quasilinear problem \begin{equation*} \left\{ \begin{array}[c]{ll} -Δu - Δ(u^2)u = |u|^{p-2}u & \mbox{ in } Ωu= 0 &\mbox{ on } \partialΩ, \end{array} \right. \end{equation*} where $Ω$ is a smooth and bounded domain in $\mathbb R^{N},N\geq3$. More specifically we prove that, for $p$ near the critical exponent $22^{*}=4N/(N-2)$, the number of positive solutions is estimated below by topological invariants of the domain $Ω$: the Ljusternick-Schnirelmann category and the Poincaré polynomial.
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 provide a method to study critical points of strongly indefinite functionals on vector bundles. We focus mainly on energy functionals coupled with a fermionic part, that is with a Dirac-type operator. We consider the cases of the perturbed Dirac-Geodesics problem and the Yang-Mills-Dirac type equation in dimension two.
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 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.
In this paper we consider the existence and multiplicity of weak solutions for the following class of fractional elliptic problem \begin{equation}\label{00} \left\{\begin{aligned} (-Δ)^{\frac{1}{2}}u + u &= Q(x)f(u)\;\;\mbox{in}\;\;\R \setminus (a,b)\\ \mathcal{N}_{1/2}u(x) &= 0\;\;\mbox{in}\;\;(a,b), \end{aligned} \right. \end{equation} where $a,b\in \R$ with $a<b$, $(-Δ)^{\frac{1}{2}}$ denotes the fractional Laplacian operator and $\mathcal{N}_s$ is the nonlocal operator that describes the Neumann boundary condition, which is given by $$ \mathcal{N}_{1/2}u(x) = \frac{1}π \int_{\R\setminus (a…
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…
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…
We shall prove a multiplicity result for semilinear elliptic problems with a super-critical nonlinearity of the form, \begin{equation}\label{con-c} \left \{ \begin{array}{ll} -Δu =|u|^{p-2} u+μ|u|^{q-2}u, & x \in Ω\\ u=0, & x \in \partial Ω\end{array} \right. \end{equation} where $Ω\subset \mathbb{R}^n$ is a bounded domain with $C^2$-boundary and $1<q< 2<p.$ As a consequence of our results we shall show that, for each $p>2$, there exists $μ^*>0$ such that for each $μ\in (0, μ^*)$ this problem has a sequence of solutions with a negative energy. This result was already known for the subcritical …
This paper deals with a coupled Hartree system with Hardy-Littlewood-Sobolev critical exponent \begin{equation*} \begin{cases} -Δu+(V_1(x)+λ_1)u=μ_1(|x|^{-4}*u^{2})u+β(|x|^{-4}*v^{2})u, \ \ &x\in R^N, -Δv+(V_2(x)+λ_2)v=μ_2(|x|^{-4}*v^{2})v+β(|x|^{-4}*u^{2})v, \ \ &x\in R^N, \end{cases} \end{equation*} where $N\geq 5$, $λ_1$, $λ_2\geq 0$ with $λ_1+λ_2\neq 0$, $V_1(x), V_{2}(x)\in L^{\frac{N}{2}}(R^N)$ are nonnegative functions and $μ_1$, $μ_2$, $β$ are positive constants. Such system arises from mathematical models in Bose-Einstein condensates theory and nonlinear optics. By variational methods…
This paper deals with existence and multiplicity of positive solutions to the following class of nonlocal equations with critical nonlinearity: \begin{equation} \tag{$\mathcal E$} (-Δ)^s u = a(x) |u|^{2^*_s-2}u+f(x)\;\;\text{in}\;\mathbb{R}^{N}, \quad u \in \dot{H}^s(\mathbb{R}^{N}), \end{equation} where $s \in (0,1)$, $N>2s$, $2_s^*:=\frac{2N}{N-2s}$, $0< a\in L^\infty(\mathbb{R}^{N})$ and $f$ is a nonnegative nontrivial functional in the dual space of $\dot{H}^s$. We prove existence of a positive solution whose energy is negative. Further, under the additional assumption that $a$ is a contin…
In the present work we briefly explain how to adapt techniques already used in fractional and $p$-fractional Laplacian cases to obtain the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution, for the critical case, and the existence of infinitely many solutions, for the subcritical case, to the Choquard Logarithmic equation, $-Δ_N u + a(x)|u|^{N-2}u + λ(\ln|\cdot|\ast |u|^{N})|u|^{N-2}u = f(u) \textrm{ in } \mathbb{R}^N $, where $ a:\mathbb{R}^N \rightarrow \mathbb{R} $, $ λ>0 $, $ N \geq 3 $ and $f: \mathbb{R} \rightarrow [0, \infty) $ is conti…
In this paper we are concerned with the number of nonnegative solutions of the elliptic system $$ {array}{ll} -Δu = Q_u(u,v) + 1/2{2^*} H_u(u,v),& {in} Ω,\vdois\ -Δv = Q_v(u,v) + 1/{2^*} H_v(u,v),& {in} Ω,\vdois\ u=v=0,& {on} \partialΩ. \leqno{(P)} $$ where $Ω\subset \mathbb{R}^N$ is a bounded smooth domain, $N \geq 3$, $2^*:=2N/(N-2)$ and $Q_u, H_u$ and $Q_v$, $H_v$ are the partial derivatives of the homogeneous functions $Q,\,H \in C^1(\mathbb{R}^2_+,\mathbb{R})$, where $ \mathbb{R}^2_+ := [0,\infty) \times [0,\infty)$. In the proofs we apply variational methods and Ljusternik-Schnirelmann t…
We study a class of critical Schrödinger-Poisson system of the form \begin{equation*} \begin{cases} -Δu+λV(x)u+ϕu=μ|u|^{p-2}u+|u|^{4}u& \quad x\in \mathbb{R}^3,\\ -Δϕ=u^2&\quad x\in \mathbb{R}^3,\\ \end{cases} \end{equation*} where $λ, μ>0$ are two parameters, $p\in(4,6)$ and $V$ satisfies some potential well conditions. By using the variational arguments, we prove the existence of positive ground state solutions for $λ$ large enough and $μ>0$, and their asymptotical behavior as $λ\to\infty$. Moreover, by using Ljusternik-Schnirelmann theory, we obtain the existence of multiple positive soluti…
We deal with the following fractional Schrödinger-Poisson equation with magnetic field \begin{equation} \varepsilon^{2s}(-Δ)_{A/\varepsilon}^{s}u+V(x)u+\varepsilon^{-2t}(|x|^{2t-3}*|u|^{2})u=f(|u|^{2})u+|u|^{2^{*}_{s}-2}u \quad \mbox{in} \mathbb{R}^{3}, \nonumber \end{equation} where $\varepsilon>0$ is a small parameter, $s\in (\frac{3}{4}, 1)$, $t\in (0,1)$, $2^{*}_{s}=\frac{6}{3-2s}$ is the fractional critical exponent, $(-Δ)^{s}_{A}$ is the fractional magnetic Laplacian, $V:\mathbb{R}^{3}\rightarrow \mathbb{R}$ is a positive continuous potential, $A:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$…
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.
