The effect of a singular term in a quadratic quasi-linear problem

Abstract

For an open bounded set $\Omega \subset \mathbb {R}^N$, $N\ge 3$, $0\lvertneqq f\in L^m(\Omega )$ with $m> 1$, $0\le \mu (x)\in L^\infty (\Omega )$ and assuming in addition $\Vert \mu \Vert _\infty < \frac{N(m-1)}{N-2m}$ if $m<\frac{N}{2}$, we prove the existence of a positive solution for the singular b.v.p

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u = \lambda u + \mu (x)\,\frac{|\nabla u |^2}{u}\, + f(x), &{} \text { in } \Omega , \\ u=0, &{} \text { on } \partial \Omega , \end{array}\right. } \end{aligned}$$

provided that $\lambda <\lambda _1/(1+\Vert \mu \Vert _{ L^\infty (\Omega ) })$ (extending the previous results in [3] for $\lambda =0$). The model case $\mu (x)\equiv B<1$ is studied in more detail obtaining in addition the uniqueness (resp. nonexistence) of positive solution if the parameter $\lambda <\frac{\lambda _1}{B+1}$ (resp. $\lambda \ge \frac{\lambda _1}{B+1}$). Even more, the solutions constitute a continuum of solutions bifurcating from infinity at $\lambda =\frac{\lambda _1}{B+1}$. This is in contrast with [5], where the multiplicity of solutions of the nonsingular problem ($\frac{1}{u}$ do not appear in the equation) is deduced due to the bifurcation from infinity at $\lambda =0$.

Appendix

Lemma 5.1

The operator K is compact, i.e., if $0\le \lambda _n$ is convergent to $\lambda$ and $w_n$ is $L^\infty$-weakly convergent to w, then the sequence of the unique solution $u_n=K_{\lambda _n}(w_n)$ of

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u_n = B\, \displaystyle \frac{|\nabla u_n |^2}{u_n} +\lambda _n w_n^+ + f(x), &{}\text { in } \Omega , \\ u_n>0, &{}\text { in } \Omega , \\ u_n=0, &{}\text { on } \partial \Omega , \end{array}\right. } \end{aligned}$$

(5.1)

is strongly convergent in $L^\infty (\Omega )$ to the unique solution $u=K_{\lambda }(w)$ of

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u = B\, \displaystyle \frac{|\nabla u |^2}{u} + \lambda w^+ + f(x), &{}\text { in } \Omega , \\ u>0, &{}\text { in } \Omega , \\ u=0, &{}\text { on } \partial \Omega . \end{array}\right. } \end{aligned}$$

(5.2)

Proof

First, we observe that the sequence $\{\lambda _n\,w_n^+\}$ is bounded in $L^\infty (\Omega )$, and thus, there exists a positive constant C, such that

$$\begin{aligned} 0\le |\lambda _n|\,w_n^+ + f \le C+f. \end{aligned}$$

(5.3)

Next, we divide the proof into several steps.

Step 1. The sequence $\{u_n\}$ is bounded in $W_0^{1,2}(\Omega )$ Indeed, thanks to Remark 4.1, it is possible to take $u_n$ as a test function in the weak formulation of (5.1) to obtain, using (5.3), that

$$\begin{aligned} (1-B)\int _{\Omega }|\nabla u_n|^2 \le \int _{\Omega } (C + f) u_n, \end{aligned}$$

which gives us the result after applying Sobolev and Hölder inequalities.

Consequently, there exists a function $u\in W_0^{1,2}(\Omega )$, such that, up to a subsequence, $u_n$ converges to u weakly in $W_0^{1,2}(\Omega )$.

Step 2. The sequence $\{u_n\}$ is bounded in $L^\infty (\Omega )$ Indeed, one can take $G_k(u_n)$ as a test function (with $k\ge 1$) in the weak formulation of (5.1) to obtain, using $|G_k(u_n)| \le |u_n|$ and (5.3), that

$$\begin{aligned} \big ( 1-B\big ) \int _{\Omega } |\nabla G_k(u_n)|^2 \le \int _{\{u_n>k\}} (C+f)\,G_k(u_n). \end{aligned}$$

Since $m > N/2$ and $0<B<1$, one can follow the arguments of Ladyzhenskaya–Ural’tseva ([15]) to deduce the result thanks to Step 1.

As a consequence, the function u given by Step 1 belongs to $L^\infty (\Omega )$.

Step 3. The sequence $\{u_n\}$ is bounded in $C^{0,\alpha }(\overline{\Omega })$ for some $\alpha \in (0,1).$ To do it, we consider a function $\xi \in C_c^\infty (\Omega )$ with $0\le \xi (x) \le 1$ for all $x\in \Omega$ and compact support in a ball $B_\rho$ of radius $\rho$. Given a solution $u_n$ of (5.1), we take $\varphi = \xi ^2 G_k(u_n)$ as a test function in the weak formulation of (5.1). Thanks to the smoothness of $\Omega$, we follow the same arguments of Theorem 1.1 of Chapter 4 in [15] to deduce $u_n\in C^{0,\alpha }(\overline{\Omega })$ and $\Vert u_n\Vert _{C^{0,\alpha }(\overline{\Omega })} \le C(\Vert u_n\Vert _{\infty },f,B,\Omega )$. Therefore, using Step 2, we deduce the boundedness of $\{u_n\}$ in $C^{0,\alpha }(\overline{\Omega })$. Since, by Ascoli–Arzela’s theorem, the embedding $C^{0,\alpha }(\overline{\Omega }) \hookrightarrow C^0(\overline{\Omega })$ is compact, then the sequence $\{u_n\}$ strongly converges in $L^\infty (\Omega )$ to the function u given by Step 1.

Step 4. The function u given by Step 1 is a solution of (5.2) Following the arguments of [3] and using the above a priori estimates, we can pass to the limit in the weak formulation of the problem (5.1) to prove that u is a solution of (5.2). $\square$