Patrick Winkert, Rico Zacher
In this paper we study elliptic equations with a nonlinear conormal derivative boundary condition involving nonstandard growth terms. By means of the localization method and De Giorgi's iteration technique we derive global a priori bounds for weak solutions of such problems.
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It is shown by means of reiterated two-scale convergence in the Sobolev-Orlicz setting, that the sequence of solutions of a class of highly oscillatory problems involving nonlinear elliptic operators with nonstandard growth, converges to a solution of a suitable homogeneous nonlinear elliptic equation associated to an operator with nonstandard growth.
Lipschitz regularity for a priori bounded minimizers of integral functionals with nonstandard growth
该研究在增长指数与椭圆性指数之差满足严格界的条件下，建立了满足非标准增长条件的非自治积分泛函先验有界局部极小元的Lipschitz正则性。
In this paper we study a rather wide class of quasilinear parabolic problems with nonlinear boundary condition and nonstandard growth terms. It includes the important case of equations with a $p(t,x)$-Laplacian. By means of the localization method and De Giorgi's iteration technique we derive global a priori bounds for weak solutions of such problems. Our results seem to be new even in the constant exponent case.
We study solutions and supersolutions of homogeneous and nonhomogeneous $\mathcal{A}$-harmonic equations with nonstandard growth in $\mathbb{R}^n$. Various Liouville-type theorems and nonexistence results are proved. The discussion is illustrated by a number of examples.
Existence of very weak solutions to nonlinear elliptic equation with nonstandard growth and global weighted gradient estimates
本文研究具有非标准增长的一般拟线性椭圆方程，通过建立极弱解的全局加权梯度估计，结合 Muckenhoupt 权的反 Hölder 类与 Orlicz 空间中的广义加权 div-curl 引理，证明了该问题极弱解的存在性。
We study the elliptic inclusion given in the following divergence form \begin{align*} & -\mathrm{div}\, A(x,\nabla u) \ni f\quad \mathrm{in}\quad Ω, & u=0\quad \mathrm{on}\quad \partial Ω. \end{align*} As we assume that $f\in L^1(Ω)$, the solutions to the above problem are understood in the renormalized sense. We also assume nonstandard, possibly nonpolynomial, heterogeneous and anisotropic growth and coercivity conditions on the maximally monotone multifunction $A$ which necessitates the use of the nonseparable and nonreflexive Musielak--Orlicz spaces. We prove the existence and uniqueness of…
We obtain new partial Hölder continuity results for solutions to divergence form elliptic systems with discontinuous coefficients, obeying $p(x)$-type nonstandard growth conditions. By an application of the method of $\mathcal{A}$-harmonic approximation, we are able to allow for both VMO-discontinuities in the coefficients, and the minimal $\log$-Hölder regularity assumption on the exponent function. In doing so, we recover a local version of the quantisation phenomenon characteristic continuous coefficient case.
In this paper, we obtain $C^{1,α}$ estimates for weak solutions of certain quasilinear parabolic equations satisfying nonstandard growth conditions, the prototype examples being $$u_t - \text{div} (|\nabla u|^{p-2} \nabla u + a(t)|\nabla u|^{q-2} \nabla u) = 0,$$ $$u_t - \text{div} (|\nabla u|^{p(t)-2} \nabla u) = 0.$$ under the assumption that the solutions a priori have bounded gradient. We build on the recently developed scaling and covering argument which allows us to consider the singular and degenerate cases in a uniform manner and with minimal regularity requirements on the phase switch…
We study asymptotic behavior of sub-solutions to non-uniformly elliptic equations with nonstandard growth. In particular, Harnack type inequalities are proved. Our approach gives new results for the cases with (p,q) nonlinearity and generalized Orlicz growth.
Parabolic De Giorgi classes with doubly nonlinear, nonstandard growth: local boundedness under exact integrability assumptions
该研究定义了含非平衡能量估计的抛物型De Giorgi函数类$\mathcal{PDG}$，证明其在奇异抛物算子的三类增长条件下局部有界，次临界情形的结果为全新成果，无需额外可积性条件。
Harnack inequality for anisotropic fully nonlinear equations with nonstandard growth
针对具有非标准增长条件的退化完全非线性各向异性椭圆方程，通过滑动抛物面方法和构造新型屏障函数建立Harnack不等式。
We introduce elliptic and parabolic $\mathcal{B}_{1}$ classes that generalize the well-known $\mathfrak{B}_{p}$ classes of DeGiorgi, Ladyzhenskaya and Ural'tseva with $p>1$. New classes are applied to prove pointwise continuity of solutions of elliptic and parabolic equations with nonstandard growth conditions. Our considerations cover new cases of variable exponent and $(p, q)$-phase growth including the ,,singular-degenerate'' parabolic case $p<2<q$.
We obtain a comparison result for solutions to nonlinear fully anisotropic elliptic problems by means of anisotropic symmetrization. As consequence we deduce a priori estimates for norms of the relevant solutions.
We establish a gradient estimate for a very weak solution to a quasilinear elliptic equation with a nonstandard growth condition, which is a natural generalization of the $p$-Laplace equation. We investigate the maximum extent for the gradient estimate to hold without imposing any regularity assumption on the nonlinearity other than basic structure assumptions. Our results also include a higher integrability result of the gradient and the existence for the very weak solutions to such nonlinear problems.
This article sets forth results on the existence, a priori estimates and boundedness of positive solutions of a singular quasilinear systems of elliptic equations involving variable exponents. The approach is based on Schauder's fixed point Theorem. A Moser iteration procedure is also obtained for singular cooperative systems involving variable exponents establishing a priori estimates and boundedness of solutions.
A priori estimates for solutions of $g$-Laplace type problems
本文针对含 $g$-Laplace 算子的非标准增长椭圆问题，利用 Gidas-Spruck 爆破论证获得弱解先验界，并据此证明解的存在性。
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.
In this paper we show existence of solutions for some elliptic problems with nonlocal diffusion by means of nonvariational tools. Our proof is based on the use of topological degree, which requires a priori bounds for the solutions. We obtain the a priori bounds by adapting the classical scaling method of Gidas and Spruck. We also deal with problems involving gradient terms.
We provide symmetrization results in the form of mass concentration comparisons for fractional singular elliptic equations in bounded domains, coupled with homogeneous external Dirichlet conditions. Two types of comparison results are presented, depending on the summability of the right-hand side of the equation. The maximum principle arguments employed in the core of the proofs of the main results offer a nonstandard, flexible alternative to the ones described in [18, Theorem 31]. Some interesting consequences are Lp regularity results and nonlocal energy estimates for solutions.
In this paper we study strongly coupled elliptic systems in non-variational form involving fractional Laplace operators. We prove Liouville type theorems and, by mean of the blow-up method, we establish a priori bounds of positive solutions for subcritical and superlinear nonlinearities in a coupled sense. By using those latter, we then derive the existence of positive solutions through topological methods.
In this paper we present a new bootstrap procedure for elliptic systems with two unknown functions. Combining with the $L^p$-$L^q$-estimates, it yields the optimal $L^\infty$-regularity conditions for the three well-known types of weak solutions: $H_0^1$-solutions, $L^1$-solutions and $L^1_δ$-solutions. Thanks to the linear theory in $L^p_δ(Ω)$, it also yields the optimal conditions for a priori estimates for $L^1_δ$-solutions. Based on the a priori estimates, we improve known existence theorems for some classes of elliptic systems.
We consider the Dirichlet problem for positive solutions of the equation $-Δ_p (u) = f(u)$ in a convex, bounded, smooth domain $Ω\subset\R^N$, with $f$ locally Lipschitz continuous. \par We provide sufficient conditions guarantying $L^{\infty} $ a priori bounds for positive solutions of some elliptic equations involving the $p$-Laplacian and extend the class of known nonlinearities for which the solutions are $L^{\infty} $ a priori bounded. As a consequence we prove the existence of positive solutions in convex bounded domains.
In this paper, we present a bootstrap procedure for general elliptic systems with $n(\geq 3)$ components. Combining with the $L^p$-$L^q$-estimates, it yields the optimal $L^\infty$-regularity conditions for the three well-known types of weak solutions: $H_0^1$-solutions, $L^1$-solutions and $L^1_δ$-solutions. Thanks to the linear theory in $L^p_δ(Ω)$, it also yields the optimal conditions for a priori estimates for $L^1_δ$-solutions. Based on the a priori estimates, we improve known existence theorems for some classes of elliptic systems.
We describe a new method of proving a priori bounds for positive supersolutions and solutions of superlinear elliptic PDE, based on global weak Harnack inequalities and a quantitative Hopf lemma. Novel results based on the method include: (i) equations without a boundary condition on the whole boundary; (ii) equations with nonlinearities which do not have precise growth at infinity; (iii) systems of inequalities with opposite sign.
We extend some classical results dealing with boundary Harnack inequatilities to a class of quasilinear elliptic equations and derive some new estimates for solutions of such equations with an isolated singularity on the boundary of a domain.
We study local and global properties of solutions of --$Δ$u = u p + M ||u| q in a domain $Ω$ of R N , in the range min{p, q} > 1 and M $\in$ R. We prove a priori estimates and existence or non-existence of ground states.
In this paper we study quasilinear elliptic systems with nonlinear boundary condition with fully coupled perturbations even on the boundary. Under very general assumptions our main result says that each weak solution of such systems belongs to $L^\infty(\close)\times L^\infty(\close)$. The proof is based on Moser's iteration scheme. The results presented here can also be applied to elliptic systems with homogeneous Dirichlet boundary condition.
In this paper, we mainly discuss a priori bounds of the following degenerate elliptic equation, {equation}\label{000} a^{ij}(x)\partial_{ij}u+b^i(x)\partial_i u +f(x,u)=0,\text{in}Ω\subset\subset R^n, {equation} where $a^{ij}\partial_i ϕ\partial_j ϕ=0$ on $\partial Ω$, $ϕ$ is the defining function of $\partial Ω$. Imposing suitable conditions on the coefficients and $f(x,u)$, one can get the $L^\infty$-estimates of \eqref{000} via blow up method.
Local boundedness of weak solutions to elliptic equations under unbalanced Orlicz growth conditions
本文研究了在不平衡Orlicz增长条件下，非线性椭圆方程弱解的局部有界性条件，并扩展了p,q-增长方程的结果。
An interpolation approach to $L^{\infty}$ a priori estimates for elliptic problems with nonlinearity on the boundary
本文研究边界非线性次临界椭圆方程弱解的 $L^\infty$ 先验估计，通过融合 Moser 迭代、椭圆正则性和 Gagliardo–Nirenberg 插值得到显式界，并用于满足 AR 条件的问题。
We consider a two-phase elliptic-parabolic moving boundary problem modelling an evaporation front in a porous medium. Our main result is a proof of short-time existence and uniqueness of strong solutions to the corresponding nonlinear evolution problem in an $L_{p}$-setting. It relies critically on nonstandard optimal regularity results for a linear elliptic-parabolic system with dynamic boundary condition.
We study qualitative properties of positive solutions of noncooperative, possibly nonvariational, elliptic systems. We obtain new classification and Liouville type theorems in the whole Euclidean space, as well as in half-spaces, and deduce a priori estimates and existence of positive solutions for related Dirichlet problems. We significantly improve the known results for a large class of systems involving a balance between repulsive and attractive terms. This class contains systems arising in biological models of Lotka-Volterra type, in physical models of Bose-Einstein condensates and in mode…
We study nonlinear measure data elliptic problems involving the operator exposing generalized Orlicz growth. Our framework embraces reflexive Orlicz spaces, as well as natural variants of variable exponent and double-phase spaces. Approximable and renormalized solutions are proven to exist and coincide for arbitrary measure datum and to be unique when the datum is diffuse with respect to a relevant nonstandard capacity. For justifying that the class of measures is natural, a capacitary characterization of diffuse measures is provided.
In the present paper, we propose the investigation of variable-exponent, degenerate/singular elliptic equations in non-divergence form. This current endeavor parallels the by now well established theory of functionals satisfying nonstandard growth condition, which in particular encompasses problems ruled by the $p(x)$-laplacian operator. Under rather general conditions, we prove viscosity solutions to variable exponent fully nonlinear elliptic equations are locally of class $C^{1,κ}$ for a universal constant $0< κ< 1$. A key feature of our estimates is that they do not depend on the modulus of…
In this article we study local and global properties of positive solutions of $-Δ_mu=|u|^{p-1}u+M|\nabla u|^q$ in a domain $Ω$ of $\mathbb R^N$, with $m>1$, $p,q>0$ and $M\in\mathbb R$. Following some ideas used in \cite{BV,Vron1}, and by using a direct Bernstein method combined with Keller-Osserman's estimate, we obtain several a priori estimates as well as Liouville type theorems. Moreover, we prove a local Harnack inequality with the help of Serrin's classical results.
We give a simpler proof of the a priori estimates obtained in the paper by Duran, Sanmartino and Toschi for solutions of elliptic problems in weighted Sobolev norms when the weights belong to the Muckenhoupt class $A_p$. The argument is a generalization to bounded domains of the one used in $\mathbb{R}^n$ to prove the continuity of singular integral operators in weighted norms. In the case of singular integral operators it is known that the $A_p$ condition is also necessary for the continuity. We do not know whether this is also true for the a priori estimates in bounded domains but we are abl…
We consider positive solutions of $Δu-μu+Ku^{\frac{n+2}{n-2}}=0$ on $B_1$ ($n\ge 5$) where $μ$ and $K>0$ are smooth functions on $B_1$. If $K$ is very sub-harmonic at each critical point of $K$ in $B_{2/3}$ and the maximum of $u$ in $\bar B_{1/3}$ is comparable to its maximum over $\bar B_1$, then all positive solutions are uniformly bounded on $\bar B_{1/3}$. As an application, a priori estimate for solutions of equations defined on $\mathbb S^n$ is derived.
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.
Liouville theorems for scaling invariant nonlinear elliptic systems (saying that the system does not possess nontrivial entire solutions) guarantee a priori estimates of solutions of related, more general systems. Assume that $p=2q+3>1$ is Sobolev subritical, $n\le3$ and $β\in{\mathbb R}$. We first prove a Liouville theorem for the system $$\left.\begin{aligned} -Δu &=|u|^{2q+2}u+β|v|^{q+2}|u|^q u, \\ -Δv &=|v|^{2q+2}v+β|u|^{q+2}|v|^q v, \end{aligned}\ \right\} \quad\hbox{in}\quad {\mathbb R}^n,$$ in the class of radial functions $(u,v)$ such that the number of nodal domains of $u,v,u-v,u+v$ i…
Interior a priori estimate for higher order elliptic systems in Orlicz spaces
研究具有变系数Calderón-Zygmund核的奇异积分算子及其与VMO函数的交换子在Orlicz空间中的有界性，为高阶椭圆算子的先验估计和内部正则性提供泛函分析基础。
A priori estimates for solutions of degenerate fully nonlinear elliptic equations with $L^p$ data
本文研究了具有L^p数据的退化全非线性椭圆方程的粘性解的先验正则性估计，证明了当非齐次项属于L^p(p>n)时的最优内部C^{1,α}估计，在临界情况下，在Lorentz条件f∈L^{n,1}下获得对数Lipschitz连续性模，利用滑动抛物面或尖点方法开发了在合适梯度 regime 中的均匀Hölder估计，并通过修正论方法建立了一个近似引理，从而推导出相应的Schauder型估计。
Uniform a priori bounds for Slightly Subcritical Elliptic Problems
本文研究轻微次临界、轻微超线性且正则变化非线性椭圆问题正弱解的一致 L^∞ 先验界，通过加权范数估计、移动平面法、Pohozaev 恒等式及 Morrey 定理完成证明。
A blow-up approach for a priori bounds in semilinear planar elliptic systems: the Brezis-Merle critical case
本文提出了一种新的方法，用于研究半线性平面椭圆系统在临界情况下的先验界问题，并证明了正解的存在性。
We consider fully nonlinear uniformly elliptic cooperative systems with quadratic growth in the gradient, such as $$ -F_i(x, u_i, Du_i, D^2 u_i)- \langle M_i(x)D u_i, D u_i \rangle =λc_{i1}(x) u_1 + \cdots + λc_{in}(x) u_n +h_i(x), $$ for $i=1,\cdots,n$, in a bounded $C^{1,1}$ domain $Ω\subset \mathbb{R}^N$ with Dirichlet boundary conditions; here $n\geq 1$, $λ\in\mathbb{R}$, $c_{ij},\, h_i \in L^\infty(Ω)$, $c_{ij}\geq 0$, $M_i$ satisfies $0<μ_1 I\leq M_i\leq μ_2 I$, and $F_i$ is an uniformly elliptic Isaacs operator. We obtain uniform a priori bounds for systems, under a weak coupling hypoth…
Blow-up estimates and a priori bounds for the positive solutions of a class of superlinear indefinite elliptic problems
本文针对超线性不定椭圆问题，结合缩放技术与广义弱Harnack不等式得到正爆破解的新爆破估计，再借助比较结果推出经典正解的精确先验界，首次证明N≥3时位势衰减速率不影响先验界获取。
We provide a-priori $L^\infty$ bounds for positive solutions to a class of subcritical elliptic problems in bounded $C^2$ domains. Our arguments rely on the moving planes method applied on the Kelvin transform of solutions. We prove that locally the image through the inversion map of a neighborhood of the boundary contains a convex neighborhood; applying the moving planes method, we prove that the transformed functions have no extremal point in a neighborhood of the boundary of the inverted domain. Retrieving the original solution $u$, the maximum of any positive solution in the domain $\Om,$ …
We consider the semilinear elliptic equation $-L u = f(u)$ in a general smooth bounded domain $Ω\subset R^{n}$ with zero Dirichlet boundary condition, where $L$ is a uniformly elliptic operator and $f$ is a $C^{2}$ positive, nondecreasing and convex function in $[0,\infty)$ such that $\frac{f(t)}{t}\rightarrow\infty$ as $t\rightarrow\infty$. We prove that if $u$ is a positive semistable solution then for every $0\leqβ<1$ we have $$f(u)\int_{0}^{u}f(t)f"(t)~e^{2β\int_{0}^{t}\sqrt{\frac{f"(s)}{f(s)}}ds}~dt\in L^{1}(Ω),$$ by a constant independent of $u$. As we shall see, a large number of result…
We consider the $2m$-th order elliptic boundary value problem $Lu=f(x,u)$ on a bounded smooth domain $Ω$ in $R^N$ with Dirichlet boundary conditions. The operator $L$ is a uniformly elliptic operator of order $2m$. We assume that for $s\to \pm\infty$ the nonlinearity $f(x,s)$ behaves like $|s|^q$ multiplied by a continuous and positive function of $x$. Here the exponent $q$ is subcritical, i.e., $q>1$ if $N<=2m$, $1<q<\frac{N+2m}{N-2m}$ if $N>2m$. We prove a priori bounds, i.e, we show that the $L^\infty$-norm of every solution $u$ is bounded by a constant independent of $u$. The solutions are…
$W^{1,p}$ priori estimates for solutions of linear elliptic PDEs on subanalytic domains
本文研究了次解析域上 $2$ 阶线性椭圆型 PDE 的解，通过给出边界奇点切锥的显式条件，证明了 $W^{1,p}$ 先验估计，其中 $p>2$ 仅依赖于奇点几何而不依赖于解 $u$。
Local Boundedness of Local Minimizers for a Class of Nonlinear Elliptic Systems with General Growth
研究一类具有非标准增长（不满足Δ2条件）的散度型非线性椭圆系统的局部极小值，证明其局部有界性。
A priori estimates for general elliptic and parabolic boundary value problems over irregular domains
本文提出了一种统一方法，用于处理一般椭圆和抛物边值问题在不规则区域上的先验估计，涵盖多种边界条件和应用领域。
Let $Ω\subset \mathbb R^N$, $N \geq 2$, be a smooth bounded domain. We consider a boundary value problem of the form $$-Δu = c_λ(x) u + μ(x) |\nabla u|^2 + h(x), \quad u \in H^1_0(Ω)\cap L^{\infty}(Ω)$$ where $c_λ$ depends on a parameter $λ\in \mathbb R$, the coefficients $c_λ$ and $h$ belong to $L^q(Ω)$ with $q>N/2$ and $μ\in L^{\infty}(Ω)$. Under suitable assumptions, but without imposing a sign condition on any of these coefficients, we obtain an a priori upper bound on the solutions. Our proof relies on a new boundary weak Harnack inequality. This inequality, which is of independent intere…
We obtain an a-priori $W_{loc}^{1,\infty }\ ( Ω;\mathbb{R}^{m}\ ) -$bound for solutions in $Ω\subset \mathbb{R}^{n} $, $n\geq 2$, to the elliptic system \begin{equation*} \sum_{i=1}^{n}\frac{\partial }{\partial x_{i}}\ ( \frac{g_{t}\ ( x,\ |Du\ | \ ) }{\ |Du\ | } u_{x_{i}}^{α}\ ) =0,\;\;\;\;\;α=1,2,\ldots ,m, \end{equation*} where $g\ ( x,t\ ) $, $g:Ω\times \ [ 0,\infty \ ) \rightarrow \ [ 0,\infty \ ) $, is a Carathéodory function, convex and increasing with respect to the gradient variable $t\in \ [ 0,\infty \ ) $. We allow $x-$dependence, which turns out to be a relevant difference with res…
Hölder regularity for logarithmic double phase problems
研究一类具可变指数的广义对数双相方程弱解的有界性与正则性，通过一般结构假设在次临界情形下推导先验估计，利用多种技术证明全局Hölder连续性，扩展了相关正则性结果。
A Priori Regularity Estimates for Ratio of Solutions to Elliptic Equations with a Product Structure of Two-Dimensional Nodal Sets
本文针对具有Lipschitz系数的椭圆方程，在节点集包含条件下，建立了两个解之比的优化C^{1,α}先验正则性估计，特别处理了节点集为二维函数乘积结构的情形。
We study positive solutions to the fractional semi-linear elliptic equation $$ (- Δ)^σu = K(x) u^\frac{n + 2 σ}{n - 2 σ} ~~~~~~ in ~ B_2 \setminus \{ 0 \} $$ with an isolated singularity at the origin, where $K$ is a positive function on $B_2$, the punctured ball $B_2 \setminus \{ 0 \} \subset \mathbb{R}^n$ with $n \geq 2$, $σ\in (0, 1)$, and $(- Δ)^σ$ is the fractional Laplacian. In lower dimensions, we show that, for any $K \in C^1 (B_2)$, a positive solution $u$ always satisfies that $u(x) \leq C |x|^{ - (n - 2 σ)/2 }$ near the origin. In contrast, we construct positive functions $K \in C^1…
Consider the following nonlinear elliptic equation of $p(x)$-Laplacian type with nonstandard growth \begin{equation*} \left\{ \begin{aligned} &{\rm div} a(Du, x)=μ\quad &\text{in}& \quad Ω, &u=0 \quad &\text{on}& \quad \partialΩ, \end{aligned} \right. \end{equation*} where $Ω$ is a Reifenberg domain in $\mathbb{R}^n$, $μ$ is a Radon measure defined on $Ω$ with finite total mass and the nonlinearity $a: \mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}^n$ is modeled upon the $p(\cdot)$-Laplacian. We prove the estimates on weighted {\it variable exponent} Lebesgue spaces for gradients of solutions t…
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 …
We consider positive semistable solutions $u$ of $Lu+f(u)=0$ with zero Dirichlet boundary condition, where $L$ is a uniformly elliptic operator and $f\in C^2$ is a positive, nondecreasing, and convex nonlinearity which is superlinear at infinity. Under these assumptions, the boundedness of all semistable solutions is expected up to dimension $n\leq 9$, but only established for $n\leq 4$. In this paper we prove the $L^\infty$ bound up to dimension $n=5$ under the following further assumption on $f$: for every $\varepsilon>0$, there exist $T=T(\varepsilon)$ and $C=C(\varepsilon)$ such that $f'(t…
This paper is concerned with the study of multiple positive solutions to the following elliptic problem involving a nonhomogeneous operator with nonstandard growth of $p$-$q$ type and singular nonlinearities \begin{equation*} \left\{ \begin{alignedat}{2} {} - \mathcal{L}_{p,q} u & {}= λ\frac{f(u)}{u^γ}, \ u>0 && \quad\mbox{ in } \, Ω, u & {}= 0 && \quad\mbox{ on } \partialΩ, \end{alignedat} \right. \end{equation*} where $Ω$ is a bounded domain in $\mathbb{R}^N$ with $C^2$ boundary, $N \geq 1$, $λ>0$ is a real parameter, $$\mathcal{L}_{p,q} u := div(|\nabla u|^{p-2} \nabla u + |\nabla u|^{q-2} …
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