In this article using Nehari manifold method we study the multiplicity of solutions of the following nonlocal elliptic system involving variable exponents and concave-convex nonlinearities: \begin{equation*} \;\;\; \begin{array}{rl} (-Δ)_{p(\cdot)}^{s} u&=λ~ a(x)| u|^{q(x)-2}u+\frac{α(x)}{α(x)+β(x)}c(x)| u|^{α(x)-2}u| v| ^{β(x)},\hspace{2mm} x\in Ω; \\ (-Δ)_{p(\cdot)}^{s} v&=μ~ b(x)| v|^{q(x)-2}v+\frac{α(x)}{α(x)+β(x)}c(x)| v|^{α(x)-2}v| u| ^{β(x)},\hspace{2.5mm} x\in Ω; \\ u=v&=0 ,\hspace{1cm} x\in Ω^c:=\mathbb R^N\setminusΩ, \end{array} \end{equation*} where $Ω\subset\mathbb R^N,~N\geq2$ is a smooth bounded domain, $λ,μ>0$ are the parameters, $s\in(0,1),$ $p\in C(\mathbb R^N\times \mathbb R^N,(1,\infty))$ and $q,α,β\in C(\overlineΩ,(1,\infty))$ are the variable exponents and $a,b,c\in C(\overlineΩ,[0,\infty))$ are the non-negative weight functions. We show that there exists $Λ>0$ such that for all $λ+μ<Λ$, there exist two non-trivial and non-negative solutions of the above problem under some assumptions on $q,α,β$.
In this paper, we consider a fractional p-Laplacian system with both concave-convex nonlinearities and sign-changing weight functions in bounded domains. With the help of the Nehari\ manifold, we prove that the system has at least two nontrivial solutions when the pair of the parameters $(λ,μ)$ belongs to a certain subset of $\mathbb{R}^{n}$
We study the combined effect of concave and convex nonlinearities on the number of positive solutions for a fractional system involving critical Sobolev exponents. With the help of the Nehari manifold, we prove that the system admits at least two positive solutions when the pair of parameters $(λ,μ)$ belongs to a suitable subset of $\R^2$.
We study a $p$-Laplacian equation involving a parameter $λ$ and a concave-convex nonlinearity containing a weight which can change sign. By using the Nehari manifold and the fibering method, we show the existence of two positive solutions on some interval $(0,λ^*+\varepsilon)$, where $λ^*$ can be characterized variationally. We also study the asymptotic behavior of solutions when $λ\downarrow 0$.
We propose existence and multiplicity results for the system of Schrödinger equations with sign-changing nonlinearities in bounded domains or in the whole space $\mathbb{R}^N$. In the bounded domain we utilize the classical approach via the Nehari manifold, which is (under our assumptions) a differentiable manifold of class $\mathcal{C}^1$ and the Fountain theorem by Bartsch. In the space $\mathbb{R}^N$ we additionally need to assume the $\mathbb{Z}^N$-periodicity of potentials and our proofs are based on the concentration-compactness lemma by Lions and the Lusternik-Schnirelmann values.
In this article, we study the existence and multiplicity of non-negative solutions of following $p$-fractional equation: $$ \quad \left\{\begin{array}{lr}\ds \quad - 2\int_{\mb R^n}\frac{|u(y)-u(x)|^{p-2}(u(y)-u(x))}{|x-y|^{n+p\al}} dxdy = \la h(x)|u|^{q-1}u+ b(x)|u|^{r-1} u \; \text{in}\; \Om \quad \quad \quad \quad u \geq 0 \; \mbox{in}\; \Om,\quad u\in W^{\al,p}(\mb R^n), \quad \quad\quad \quad\quad u =0\quad\quad \text{on} \quad \mb R^n\setminus \Om \end{array} \right. $$ where $\Om$ is a bounded domain in $\mb R^n$, $p\geq 2$, $n> p\al$, $\al\in(0,1)$, $0< q<p-1 <r < \frac{np}{n-ps}-1$, $…
This paper is concerned with multiplicity results for parametric singular double phase problems in $\mathbb{R}^N$ via the Nehari manifold approach. It is shown that the problem under consideration has at least two nontrivial weak solutions provided the parameter is sufficiently small. The idea is to split the Nehari manifold into three disjoint parts minimizing the energy functional on two of them. The third set turns out to be the empty set for small values of the parameter.
In this paper, we deal with equations of variational form which Nahari manifolds can contain more than two different types of critical points. We introduce a method of separating critical points on the Nahari manifold, based on the use of nonlinear generalized Rayleigh quotients. The method is illustrated by establishing the existence of positive solutions, ground states and multiplicity results for a two-parameter nonlinear elliptic boundary problem with polynomial nonlinearities.
A class of generalized Schrödinger elliptic problems involving concave-convex and other types of nonlinearities is studied. A reasonable overview about the set of solutions is provided when the parameters involved in the equation assume different real values.
By using the fibering method jointly with Nehari manifold techniques, we obtain the existence of multiple solutions to a fractional $p$-Laplacian system involving critical concave-convex nonlinearities provided that a suitable smallness condition on the parameters involved is assumed. The result is obtained despite there is no general classification for the optimizers of the critical fractional Sobolev embedding.
In this paper, we prove multiplicity of solutions for a class of quasilinear problems in $ \mathbb{R}^{N} $ involving variable exponents and nonlinearities of concave-convex type. The main tools used are variational methods, more precisely, Ekeland's variational principle and Nehari manifolds.
We consider a nonlinear Dirichlet problem driven by the $(p,q)$-Laplacian with $1<q<p$. The reaction is parametric and exhibits the competing effects of a singular term and of concave and convex nonlinearities. We are looking for positive solutions and prove a bifurcation-type theorem describing in a precise way the set of positive solutions as the parameter varies. Moreover, we show the existence of a minimal positive solution and we study it as a function of the parameter.
It is established existence of ground and bound state solutions for Choquard equation considering concave-convex nonlinearities in the following form $$ \begin{array}{rcl} -Δu +V(x) u &=& (I_α* |u|^p)|u|^{p-2}u+ λ|u|^{q-2}u, \, u \in H^1(\mathbb{R}^{N}), \end{array} $$ where $λ> 0, N \geq 3, α\in (0, N)$. The potential $V$ is a continuous function and $I_α$ denotes the standard Riesz potential. Assume also that $1 < q < 2,~2_α < p < 2^*_α$ where $2_α=(N+α)/N$, $2_α=(N+α)/(N-2)$. Our main contribution is to consider a specific condition on the parameter $λ> 0$ taking into account the nonlinear …
In this paper, by utilizing a newly established variational principle on convex sets, we provide an existence and multiplicity result for a class of semilinear elliptic problems defined on the whole $\mathbb R^N$ with nonlinearities involving sublinear and superlinear terms. We shall impose no growth restriction on the nonlinear term and consequently our problem can be super-critical by means of Sobolev spaces.
We obtain new concavity results, up to a suitable transformation, for a class of quasi-linear equations in a convex domain involving the $p$-Laplace operator and a general nonlinearity satisfying concavity type assumptions. This provides an extension of results previously known in the literature only for the torsion and the eigenfunction equations. In the semilinear case $p = 2$ the results are already new since they include new admissible nonlinearities.
It is established existence, multiplicity and asymptotic behavior of positive solutions for a quasilinear elliptic problem driven by the $Φ$-Laplacian operator. One of these solutions is obtained as ground state solution by applying the well known Nehari method. The semilinear term in the quasilinear equation is a concave-convex function which presents a critical behavior at infinity. The concentration compactness principle is used in order to recover the compactness required in variational methods.
We look for ground state solutions to the following nonlinear Schrödinger equation $$-Δu + V(x)u = f(x,u)-Γ(x)|u|^{q-2}u\hbox{ on }\mathbb{R}^N,$$ where $V=V_{per}+V_{loc}\in L^{\infty}(\mathbb{R}^N)$ is the sum of a periodic potential $V_{per}$ and a localized potential $V_{loc}$, $Γ\in L^{\infty}(\mathbb{R}^N)$ is periodic and $Γ(x)\geq 0$ for a.e. $x\in\mathbb{R}^N$ and $2\leq q<2^*$. We assume that $\infσ(-Δ+V)>0$, where $σ(-Δ+V)$ stands for the spectrum of $-Δ+V$ and $f$ has the subcritical growth but higher than $Γ(x)|u|^{q-2}u$, however the nonlinearity $f(x,u)-Γ(x)|u|^{q-2}u$ may chang…
In this article we study the existence of sign changing solution of the following p-fractional problem with concave-critical nonlinearities: \begin{eqnarray*} (-Δ)^s_pu &=& μ|u|^{q-1}u + |u|^{p^*_s-2}u \quad\mbox{in}\quad Ω, u&=&0\quad\mbox{in}\quad\mathbb{R}^N\setminusΩ, \end{eqnarray*} where $s\in(0,1)$ and $p\geq 2$ are fixed parameters, $0<q<p-1$, $μ\in\mathbb{R}^+$ and $p_s^*=\frac{Np}{N-ps}$. $Ω$ is an open, bounded domain in $\mathbb{R}^N$ with smooth boundary with $N>ps$ .
In this paper, we study the existence of multiple positive solutions for a class of fractional Schrödinger-Poisson systems involving sign-changing potential and critical nonlinearities on an unbounded domain. With the help of Nehari manifold and Ljusternik-Schnirelmann category, we investigate how the coefficient $g(x)$ of the critical nonlinearity affects the number of positive solutions. Moreover, we present a novel relationship between the number of positive solutions and the category of the global maximum set of $g(x)$.
We study the existence of least energy sign-changing solution for the fractional equation $(-Δ)^{s} u=|u|^{2_{s}^{*}-2}u+λf(x,u)$ in a smooth bounded domain $Ω$ of $\mathbb{R}^{N},$ $u=0$ in $\mathbb{R}^{N}\setminus Ω$, where $s\in (0,1)$ and $2_{s}^{*}$ is the fractional critical Sobolev exponent. The proof is based on constrained minimization in a subset of Nehari manifold, containing all the possible sign-changing solutions of the equation.
In this paper, we consider the following quasilinear Schrödinger equation \begin{align*} -Δu-uΔ(u^{2})=k(x)\left\vert u\right\vert ^{q-2}u-h(x)\left\vert u\right\vert ^{s-2}u\text{, }u\in D^{1,2}(\mathbb{R}^{N})\text{,} \end{align*} where $1<q<2<s<+\infty$. Unlike most results in the literature, the exponent $s$ here is allowed to be supercritical $s>2\cdot2^{\ast}$. By taking advantage of geometric properties of a nonlinear transformation $f$ and a variant of Clark's theorem, we get a sequence of solutions with negative energy in a space smaller than $D^{1,2}(\mathbb{R}^{N})$. Nonnegative sol…
In this paper we prove the existence of infinitely many nontrivial solutions of the following equations driven by a nonlocal integro-differential operator $L_K$ with concave-convex nonlinearities and homogeneous Dirichlet boundary conditions \begin{eqnarray*} \mathcal{L}_{K} u + μ\, |u|^{q-1}u + λ\,|u|^{p-1}u &=& 0 \quad\text{in}\quad Ω, \\[2mm] u&=&0 \quad\mbox{in}\quad\mathbb{R}^N\setminusΩ, \end{eqnarray*} where $Ω$ is a smooth bounded domain in $\mathbb{R}^N$, $N>2s$, $s\in(0, 1)$, $0<q<1<p\leq \frac{N+2s}{N-2s}$. Moreover, when $L_K$ reduces to the fractional laplacian operator $-(-Δ)^s $…
In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave-convex nonlinearities: $$({-}{ Δ})^{\fracα{2}}u- γ\frac{u}{|x|^α}= λf(x) |u|^{q - 2} u + g(x) {\frac{|u|^{p-2}u}{|x|^s}} \ \text{ in } {Ω,} \quad \text{ with Dirichlet boundary condition } u = 0 \ \text{ in } \mathbb{R}^n \setminus Ω,$$ where $Ω\subset \mathbb{R}^n$ is a smooth bounded domain in $\mathbb{R}^n$ containing $0$ in its interior, and $f,g \in C(\overlineΩ)$ with $f^+,g^+ \not\equiv 0$ which may change sign in $\overlineΩ.$ We use the var…
We are looking for solutions to nonlinear Schrödinger-type equations of the form $$ (-Δ)^{α/ 2} u (x) + V(x) u(x) = h (x,u(x)), \quad x \in \mathbb{R}^N, $$ where $V : \mathbb{R}^N \rightarrow \mathbb{R}$ is an external potential (bounded, unbounded or singular) and the nonlinearity $h : \mathbb{R}^N \times \mathbb{R} \rightarrow \mathbb{R}$ may be sign-changing. The aim of this paper is to give an introduction to the problem and the variational approach based on the Nehari manifold technique, and to survey recent results concerning the existence of ground and bound states.
In this work we study the following fractional critical problem $$ (P_λ)=\left\{\begin{array}{ll} (-Δ)^s u=λu^{q} + u^{2^*_{s}-1}, \quad u{>}0 & \mbox{in} Ω\\ u=0 & \mbox{in} \RR^n\setminus Ω\,, \end{array}\right. $$ where $Ω\subset \mathbb{R}^n$ is a regular bounded domain, $λ>0$, $0<s<1$ and $n>2s$. Here $(-Δ)^s$ denotes the fractional Laplace operator defined, up to a normalization factor, by $$ -(-Δ)^s u(x)={\rm P. V.} \int_{\RR^n}\frac{u(x+y)+u(x-y)-2u(x)}{|y|^{n+2s}}\,dy, \quad x\in \RR^n. $$ Our main results show the existence and multiplicity of solutions to problem $(P_λ)$ for differe…
We consider a nonlinear Dirichlet problem driven by a variable exponent $p$-Laplacian plus an indefinite potential term. The reaction has the competing effects of a parametric concave (sublinear) term and of a convex (superlinear) perturbation (an anisotropic concave-convex problem). We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the positive parameter $λ$ varies. Also, we prove the existence of minimal positive solutions.
In this paper, we consider the multiplicity of solutions for a class of Kirchhoff type problems with sub-linear and critical terms on an unbounded domain. With the aid of Ekeland's variational principle and the concentration compactness principle we prove that the Kirchhoff problem has at least two solutions.
In this paper we study the existence of multiple nontrivial positive weak solutions to the following system of problems. \begin{align*} \begin{split} -Δ_{p}u-Δ_q u &= λf(x)|u|^{r-2}u+ν\frac{1-α}{2-α-β}h(x) |u|^{-α}|v|^{1-β}\,\,\mbox{in}\,\,Ω,\\ -Δ_{p}v-Δ_q v &= μg(x)|v|^{r-2}v+ν\frac{1-β}{2-α-β}h(x) |u|^{1-α}|v|^{-β}\,\,\mbox{in}\,\,Ω,\\ u,v&>0\,\,\mbox{in}\,\,Ω,\\ u= v &= 0\,\, \mbox{on}\,\, \partialΩ\end{split} \end{align*} where (C):~$0<α<1,\;0<β<1,$ $2-α-β<q<\frac{N(p-1)}{N-p}<p<r<p^*$, with $p^*=\frac{Np}{N-p}$. We will guarantee the existence of a solution in the Nehari manifold. Further…
In this paper, we consider the singularly perturbed fractional Schrödinger equation \begin{equation*} ε^{2α}(-Δ)^αu+V(x)u=f(u),\quad x\in \mathbb{R}^N, \end{equation*} where $ε>0$ is a small parameter, $α\in(0,1)$, $(-Δ)^α$ is the fractional Laplacian operator of order $α$, $V$ possesses global minimum points, and $f$ is asymptotically linear at infinity. We investigate the relationship between the number of positive solutions and the topology of the set where the potential $V$ attains its global minimum. We also construct multiple concentrating solutions if $V$ has several strict global minim…
In this article, we study the existence of non-negative solutions of the class of non-local problem of $n$-Kirchhoff type $$ \left\{ \begin{array}{lr} \quad - m(\int_Ω|\nabla u|^n)Δ_n u = f(x,u) \; \text{in}\; Ω,\quad u =0\quad\text{on} \quad \partial Ω, \end{array} \right.$$ where $Ω\subset \mathbf{R}^n$ is a bounded domain with smooth boundary, $n\geq 2$ and $f$ behaves like $e^{|u|^{\frac{n}{n-1}}}$ as $|u|\to\infty$. Moreover, by minimization on the suitable subset of the Nehari manifold, we study the existence and multiplicity of solutions, when $f(x,t)$ is concave near $t=0$ and convex a…
We look for solutions to a fractional Schrödinger equation of the following form $$ (-Δ)^{α/ 2} u + \left( V(x) - \fracμ{|x|^α} \right) u = f(x,u)-K(x)|u|^{q-2}u\hbox{ on }\mathbb{R}^N \setminus \{0\}, $$ where $V$ is bounded and close-to-periodic potential and $- \fracμ{|x|^α}$ is a Hardy-type potential. We assume that $V$ is positive and $f$ has the subcritical growth but not higher than $|u|^{q-2}u$. If $μ$ is positive and small enough we find a ground state solution, i.e. a critical point of the energy being minimizer on the Nehari manifold. If $μ$ is negative we show that there is no grou…
The purpose of this paper is to prove some existence and non-existence theorems for the nonlinear elliptic problems of the form -Δ_{p}u=λk(x)u^{q}\pmh(x)u^{σ} if x\inΩ, subject to the Dirichlet conditions u_{1}=u_{2}=0 on \partialΩ. In the proofs of our results we use the sub-super solutions method and variational arguments. Related results as obtained here have been established in [Z. Guo and Z. Zhang, W^{1,p} versus C^{1} local minimizers and multiplicity results for quasilinear elliptic equations, Journal of Mathematical Analysis and Applications, Volume 286, Issue 1, Pages 32-50, 1 October…
In this paper, we study a class of fractional Schrödinger equations involving logarithmic and critical nonlinearities on an unbounded domain, and show that such an equation with positive or sign-changing weight potentials admits at least one positive ground state solution and the associated energy is positive (or negative). By applying the Nehari manifold method and Ljusternik-Schnirelmann category, we deeply investigate how the weight potential affects the multiplicity of positive solutions, and obtain the relationship between the number of positive solutions and the category of some sets rel…
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 article, we study the existence/multiplicity results for the following variable order nonlocal Choquard problem with variable exponents \begin{equation*} \begin{array}{rl} (-Δ)_{p(\cdot)}^{s(\cdot)}u(x)&=λ|u(x)|^{α(x)-2}u(x)+\left(\DD\int_Ω\frac{F(y,u(y))}{|x-y|^{μ(x,y)}}dy\right)f(x,u(x)),\\ &~\hspace{6cm} x\in Ω, \\ u(x)&=0 ,\hspace{20mm} x\in Ω^c:=\mathbb R^N\setminusΩ, \end{array} \end{equation*} where $\Om\subset\mathbb R^N$ is a smooth and bounded domain, $N\geq 2$, $p,s,μ$ and $α$ are continuous functions on $\mathbb R^N\times\mathbb R^N$ and $f(x,t)$ is continuous function with…
In this paper we consider a class of critical concave convex Ambrosetti-Prodi type problems for the fractional $p$-Laplacian operator. By applying the Linking Theorem and the Mountain Pass Theorem as well, the interaction of the nonlinearities with the first eigenvalue of fractional $p$-Laplacian will be used to prove existence and multiplicity of solutions.
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.
\noi In this article, we study the existence of non-negative solutions of the following polyharmonic Kirchhoff type problem with critical singular exponential nolinearity $$ \quad \left\{ \begin{array}{lr} \quad -M\left(\displaystyle\int_Ω|\nabla^m u|^{\frac{n}{m}}dx\right)Δ_{\frac{n}{m}}^{m} u = \frac{f(x,u)}{|x|^α} \; \text{in}\; \Om{,} \quad \quad u = \nabla u=\cdot\cdot\cdot= {\nabla}^{m-1} u=0 \quad \text{on} \quad \partial \Om{,} \end{array} \right. $$ where $\Om\subset \mb R^n$ is a bounded domain with smooth boundary, $n\geq 2m\geq 2$ and $f(x,u)$ behaves like $e^{|u|^{\frac{n}{n-m}}}$…
We consider the parabolic, initial value problem $$ v_t =Δ_p(v)+λg(x,v)ϕ_p(v), \quad \text{in $Ω\times (0,\infty),$} $$ \[ v =0, \text{in $\partialΩ\times (0,\infty),$}\tag{IVP} v =v_0\ge0, \text{in $Ω\times \{0\},$} \] where $Ω$ is a bounded domain in ${\mathbb R}^N$, for some integer $N\ge1$, with smooth boundary $\partialΩ$, $ϕ_p(s):=|s|^{p-1} {\rm sgn}s$, $s\in{\mathbb R}$, $Δ_p$ denotes the $p$-Laplacian, with $p>\max\{2,N\}$, $v_0\in C^0(\overlineΩ)$, and $λ>0$. The function $g:\overline{Ω} \times [0,\infty)\to(0,\infty)$ is $C^0$ and, for each $x\in\overline{Ω}$, the function $g(x,\cdot…
In this paper, we consider a non-local diffusion equation involving the fractional $p(x)$-Laplacian with nonlinearities of variable exponent type. Employing the sub-differential approach we establish the existence of local solutions. By combining the potential well theory with the Nehari manifold, we obtain the existence of global solutions and finite time blow-up of solutions. Moreover, we study the asymptotic stability of global solutions as time goes to infinity in some variable exponent Lebesgue spaces.
In this work, we study the higher order Kirchhoff type Choquard equation $(KC)$ involving a critical exponential non-linearity and singular weights. We prove the existence of solution to $(KC)$ using Mountain pass Lemma in light of Moser-Trudinger and singular Adams-Moser inequalities. In the second part of the paper, using the Nehari manifold technique and minimization over its suitable subsets, we prove the existence of at least two solutions to the Kirchhoff type Choquard equation $(\mathcal{P_{\la,M}})$ involving convex-concave type non-linearity.
