The first two authors [Proc. Lond. Math. Soc. (3) {\bf 114}(1):1--34, 2017] classified the behaviour near zero for all positive solutions of the perturbed elliptic equation with a critical Hardy--Sobolev growth
$$-Δu=|x|^{-s} u^{2^\star(s)-1} -μu^q \hbox{ in }B\setminus\{0\},$$ where $B$ denotes the open unit ball centred at $0$ in $\mathbb{R}^n$ for $n\geq 3$, $s\in (0,2)$, $2^\star(s):=2(n-s)/(n-2)$, $μ>0$ and $q>1$. For $q\in (1,2^\star-1)$ with $2^\star=2n/(n-2)$, it was shown in the op. cit. that the positive solutions with a non-removable singularity at $0$ could exhibit up to three different singular profiles, although their existence was left open. In the present paper, we settle this question for all three singular profiles in the maximal possible range. As an important novelty for $μ>0$, we prove that for every $q\in (2^\star(s) -1,2^\star-1)$ there exist infinitely many positive solutions satisfying $|x|^{s/(q-2^\star(s)+1)}u(x)\to μ^{-1/(q-2^\star(s)+1)}$ as $|x|\to 0$, using a dynamical system approach. Moreover, we show that there exists a positive singular solution with $\liminf_{|x|\to 0} |x|^{(n-2)/2} u(x)=0$ and
$\limsup_{|x|\to 0} |x|^{(n-2)/2} u(x)\in (0,\infty)$ if (and only if) $q\in (2^\star-2,2^\star-1)$.
Existence of sign-changing solutions for a logarithmic weighted Kirchhoff problem in the whole of $\mathbb{R}^{N}$ with exponential growth non-linearity
In this paper, we deal with a fractional elliptic equation with critical Sobolev nonlinearity and Hardy term $$ (-Δ)^α u-μ\frac{u}{|x|^{2α}}+a(x) u=|u|^{2^*-2}u+k(x)|u|^{q-2}u$$ $$ u\,\in\,H^α({\mathbb R}^N),$$ where $2<q< 2^*$, $0<α<1$, $N>4α$, $2^*=2N/(N-2α)$ is the critical Sobolev exponent, $a(x),k(x)\in C({\mathbb R}^N)$. Through a compactness analysis of the functional associated to $(*)$, we obtain the existence of positive solutions for $(*)$ under certain assumptions on $a(x),k(x)$.
In this work we consider the following class of elliptic problems $$- Δ_A u + u = a(x) |u|^{q-2}u+b(x) |u|^{p-2}u , \mbox{ in } \mathbb{R}^N, $$ $u\in H^1_A (\mathbb{R}^N)$, with $2<q<p<2^*= \frac{2N}{N-2}$, $a(x)$ and $b(x)$ are functions that can change signal and satisfy some additional conditions; $u \in H^1_A(\mathbb{R}^N)$ and $A:\mathbb{R}^N \rightarrow \mathbb{R}^N$ is a magnetic potential. Also using the Nehari method in combination with other complementary arguments, we discuss the existence of infinite solutions to the problem in question, varying the assumptions about the weight fu…
Given a linear elliptic equation $\sum a_{ij} u_{ij} =0$ in $\mathbb{R}^3$, it is a classical problem to determine if its degree-one homogeneous solutions $u$ are linear. The answer is negative in general, by a construction of Martinez-Maure. In contrast, the answer is affirmative in the uniformly elliptic case, by a theorem of Han, Nadirashvili and Yuan, and it is a known open problem to determine the degenerate ellipticity condition on $(a_{ij})$ under which this theorem still holds. In this paper we solve this problem. We prove the linearity of $u$ under the following degenerate ellipticity…
In this paper we prove the monotonicity of positive solutions to $ -Δ_p u = f(u) $ in half-spaces under zero Dirichlet boundary conditions, for $(2N+2)/(N+2) < p < 2$ and for a general class of regular changing-sign nonlinearities $f$. The techniques used in the proof of the main result are based on a fine use of comparison and maximum principles and on an adaptation of the celebrated moving plane method to quasilinear elliptic equations in unbounded domains.
We prove sharp pointwise blow-up estimates for finite-energy sign-changing solutions of critical equations of Schrödinger-Yamabe type on a closed Riemannian manifold $(M,g)$ of dimension $n \ge 3$. This is a generalisation of the so-called $C^0$-theory for positive solutions of Schrödinger-Yamabe type equations. To deal with the sign-changing case we develop a method of proof that combines an \emph{a priori} bubble-tree analysis with a finite-dimensional reduction, and reduces the proof to obtaining sharp \emph{a priori} blow-up estimates for a linear problem.
We study the asymptotic and qualitative properties of least energy radial sign-changing solutions to fractional semilinear elliptic problems of the form \[ \begin{cases} (-Δ)^s u = |u|^{2^*_s-2-\varepsilon}u &\text{in } B_R, \\ u = 0 &\text{in }\mathbb{R}^n \setminus B_R, \end{cases} \] where $s \in (0,1)$, $(-Δ)^s$ is the s-Laplacian, $B_R$ is a ball of $\mathbb{R}^n$, $2^*_s := \frac{2n}{n-2s}$ is the critical Sobolev exponent and $\varepsilon>0$ is a small parameter. We prove that such solutions have the limit profile of a "tower of bubbles", as $ \varepsilon \to 0^+$, i.e. the positive and…
In this paper, we investigate the existence of nontrivial weak solutions to a class of elliptic equations ($\mathscr{P}$) involving a general nonlocal integrodifferential operator $\mathscr{L}_{\mathcal{A}K}$, two real parameters, and two weight functions, which can be sign-changing. Considering different situations concerning the growth of the nonlinearities involved in the problem ($\mathscr{P}$), we prove the existence of two nontrivial distinct solutions and the existence of a continuous family of eigenvalues. The proofs of the main results are based on ground state solutions using the Neh…
In this note we prove the Payne-type conjecture about the behaviour of the nodal set of least energy sign-changing solutions for the equation $-Δ_p u = f(u)$ in bounded Steiner symmetric domains $Ω\subset \mathbb{R}^N$ under the zero Dirichlet boundary conditions. The nonlinearity $f$ is assumed to be either superlinear or resonant. In the latter case, least energy sign-changing solutions are second eigenfunctions of the zero Dirichlet $p$-Laplacian in $Ω$. We show that the nodal set of any least energy sign-changing solution intersects the boundary of $Ω$. The proof is based on a moving polar…
We are concerned on the fourth-order elliptic equation \begin{equation}\tag{$P_λ$} \left\{ \begin{array}[c]{ll} Δ^2 u- Δu + V(x)u -λΔ[ρ(u^2)]ρ'(u^2)u= f(u)\, \, \mbox{in} \, \, \mathbb{R}^N, & u\in W^{2,2}(\mathbb{R}^N), \end{array} \right. \end{equation} where $Δ^2 = Δ(Δ)$ is the biharmonic operator, $3\leq N\leq 6$, the radially symmetric potential $V$ may change sign and $\inf_{\mathbb{R}^N}V(x)=-\infty$ is allowed. If $f$ satisfies a type of nonquadracity and monotonicity conditions and $ρ$ is a suitable smooth function, we prove, via variational approach, the existence of a radially symme…
We consider the fractional elliptic inequality with variable-exponent nonlinearity $$ (-Δ)^{\fracα{2}} u+λ\, Δu \geq |u|^{p(x)}, \quad x\in\mathbb{R}^N, $$ where $N\geq 1$, $α\in (0,2)$, $λ\in\mathbb{R}$ is a constant, $p: \mathbb{R}^N\to (1,\infty)$ is a measurable function, and $(-Δ)^{\fracα{2}}$ is the fractional Laplacian operator of order $\fracα{2}$. A Liouville-type theorem is established for the considered problem. Namely, we obtain sufficient conditions under which the only weak solution is the trivial one. Next, we extend our study to systems of fractional elliptic inequalities with …
We investigate the regularity of semi-stable, radially symmetric, and decreasing solutions for a class of quasilinear reaction-diffusion equations in the inhomogeneous context of Riemannian manifolds. We prove uniform boundedness, Lebesgue and Sobolev estimates for this class of solutions for equations involving the p-Laplace Beltrami operator and locally Lipschitz non-linearity. We emphasize that our results do not depend on the boundary conditions and the specific form of the non-linearities and metric. Moreover, as an application, we establish regularity of the extremal solutions for equati…
We study the existence and asymptotic behavior of solutions having positive and sign-changing components to the singularly perturbed system of elliptic equations \begin{equation*} \begin{cases} -\varepsilon^2Δu_i+u_i=μ_i|u_i|^{p-2}u_i + \sum\limits_{\substack{j=1 \\ j \not=i}}^\ellλ_{ij}β_{ij}|u_j|^{α_{ij}}|u_i|^{β_{ij} -2}u_i,\\ u_i \in H^1_0(Ω), \quad u_i\neq 0, \qquad i=1,\ldots,\ell, \end{cases} \end{equation*} in a bounded domain $Ω$ in $\mathbb{R}^N$, with $N\geq 4$, $\varepsilon>0$, $μ_i>0$, $λ_{ij}=λ_{ji}<0$, $α_{ij}, β_{ij}>1$, $α_{ij}=β_{ji}$, $α_{ij} + β_{ij} = p\in (2,2^*)$, and $2…
We present new $L^\infty$ a priori estimates for weak solutions of a wide class of subcritical elliptic equations in bounded domains. No hypotheses on the sign of the solutions, neither of the non-linearities are required. This method is based in combining elliptic regularity with Gagliardo-Nirenberg or Caffarelli-Kohn-Nirenberg interpolation inequalities. Let us consider a semilinear boundary value problem $ -Δu= f(x,u),$ in $Ω,$ with Dirichlet boundary conditions, where $Ω\subset \mathbb{R}^N $, with $N> 2,$ is a bounded smooth domain, and $f$ is a subcritical Carathéodory non-linearity. We …
n this paper, we prove existence of nodal solutions for singular semilinear elliptic systems without variational structure where its both components are of sign changing. Our approach is based on sub-supersolutions method combined with perturbation arguments involving singular terms.
We consider the mixed local-nonlocal semi-linear elliptic equations driven by the superposition of Brownian and Lévy processes \begin{equation*} \left\{ \begin{array}{ll} - Δu + (-Δ)^s u = g(x,u) & \hbox{in $Ω$,} u=0 & \hbox{in $\mathbb{R}^n\backslashΩ$.} \\ \end{array} \right. \end{equation*} Under mild assumptions on the nonlinear term $g$, we show the $L^\infty$ boundedness of any weak solution (either not changing sign or sign-changing) by the Moser iteration method. Moreover, when $s\in (0, \frac{1}{2}]$, we obtain that the solution is unique and actually belongs to $C^{1,α}(\overlineΩ)$ …
Let $Ω\subset \mathbb{R}^N$, $N \geq 2$, be a smooth bounded domain. We consider the boundary value problem \begin{equation} \label{Plambda-Abstract-ch3} \tag{$P_λ$} -Δu = c_λ(x) u + μ|\nabla u|^2 + h(x)\,, \quad u \in H_0^1(Ω) \cap L^{\infty}(Ω)\,, \end{equation} where $c_λ$ and $h$ belong to $L^q(Ω)$ for some $q > N/2$, $μ$ belongs to $\mathbb{R} \setminus \{0\}$ and we write $c_λ$ under the form $c_λ:= λc_{+} - c_{-}$ with $c_{+} \gneqq 0$, $c_{-} \geq 0$, $c_{+} c_{-} \equiv 0$ and $λ\in \mathbb{R}$. Here $c_λ$ and $h$ are both allowed to change sign. As a first main result we give a neces…
In this paper we study the existence of a least energy sign-changing solution to a nonlocal elliptic PDE involving singularity by using the Nehari manifold method, the constraint variational method and Brouwer degree theory.
The present paper studies the fractional $p$-Laplacian boundary value problems with jumping nonlinearities at zero or infinity and obtain the existence of multiple solutions and sign-changing solutions by constructing the suitable pseudo-gradient vector field of the corresponding energy functional.
We investigate the possible blow-up of strong solutions to a biological network formation model originally introduced by D. Cai and D. Hu \cite{HC}. The model is represented by an initial boundary value problem for an elliptic-parabolic system with cubic non linearity. We obtain an algebraic equation for the possible blow-up time of strong solutions. The equation yields information on how various given data may contribute to the blow-up of solutions. As a by-product of our development, we establish a $W^{1,q}$ estimate for solutions to an elliptic equation which shows the explicit dependence o…
