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Register a new userInfinitely many small solutions to an elliptic PDE of variable exponent with a singular nonlinearity
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We prove the existence of infinitely many nonnegative solutions to the following nonlocal elliptic partial differential equation involving singularities begin{align} (-Delta)_{p(cdot)}^{s} u&=frac{lambda}{|u|^{gamma(x)-1}u}+f(x,u)~text{in}~Omega, onumber u&=0~text{in}~mathbb{R}^NsetminusOmega, onumber end{align} where $Omegasubsetmathbb{R}^N,, Ngeq2$ is a smooth, bounded domain, $lambda>0$, $sin (0,1)$, $gamma(x)in(0,1)$ for all $xinbar{Omega}$, $N>sp(x,y)$ for all $(x,y)inbar{Omega}timesbar{Omega}$ and $(-Delta)_{p(cdot)}^{s}$ is the fractional $p(cdot)$-Laplacian operator with variable exponent. The nonlinear function $f$ satisfies certain growth conditions. Moreover, we establish a uniform $L^{infty}(bar{Omega})$ estimate of the solution(s) by the Moser iteration technique.
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This paper is concerned with the $p(x)$-Laplacian equation of the form begin{equation}label{eq0.1} left{begin{array}{ll} -Delta_{p(x)} u=Q(x)|u|^{r(x)-2}u, &mbox{in} Omega, u=0, &mbox{on} partial Omega, end{array}right. end{equation} where $OmegasubsetR^N$ is a smooth bounded domain, $1<p^-=min_{xinoverline{Omega}}p(x)leq p(x)leqmax_{xinoverline{Omega}}p(x)=p^+<N$, $1leq r(x)<p^{*}(x)=frac{Np(x)}{N-p(x)}$, $r^-=min_{xin overline{Omega}}r(x)<p^-$, $r^+=max_{xinoverline{Omega}}r(x)>p^+$ and $Q: overline{Omega}toR$ is a nonnegative continuous function. We prove that eqref{eq0.1} has infinitely many small solutions and infinitely many large solutions by using the Clarks theorem and the symmetric mountain pass lemma.
We consider a sequence of blowup solutions of a two dimensional, second order elliptic equation with exponential nonlinearity and singular data. This equation has a rich background in physics and geometry. In a work of Bartolucci-Chen-Lin-Tarantello it is proved that the profile of the solutions differs from global solutions of a Liouville type equation only by a uniformly bounded term. The present paper improves their result and establishes an expansion of the solutions near the blowup points with a sharp error estimate.
We consider a parabolic-type PDE with a diffusion given by a fractional Laplacian operator and with a quadratic nonlinearity of the gradient of the solution, convoluted with a singular term b. Our first result is the well-posedness for this problem: We show existence and uniqueness of a (local in time) mild solution. The main result is about blow-up of said solution, and in particular we find sufficient conditions on the initial datum and on the term b to ensure blow-up of the solution in finite time.
We consider positive singular solutions to semilinear elliptic problems with possibly singular nonlinearity. We deduce symmetry and monotonicity properties of the solutions via the moving plane procedure.
We study a class of elliptic problems with homogeneous Dirichlet boundary condition and a nonlinear reaction term $f$ which is nonlocal depending on the $L^{p}$-norm of the unknown function. The nonlinearity $f$ can make the problem degenerate since it may even have multiple singularities in the nonlocal variable. We use fixed point arguments for an appropriately defined solution map to produce multiplicity of classical positive solutions with ordered norms.
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